摘要:

GENERAL DISCUSSIONS OF THE FARADAY SOCIETY Date 1907 1907 1910 1911 1912 1913 1913 1913 1914 1914 1915 1916 1916 1917 1917 1917 1918 1918 1918 1918 1919 1919 1920 1920 1920 1920 1921 1921 1921 1921 1922 1922 1923 1923 1923 1923 1923 I924 1924 1924 1924 1924 1925 1925 1926 1926 1927 1927 1927 Subject Osmotic Pressure Hydrates in Solution The Constitution of Water High Temperature Work Magnetic Properties of Alloys Colloids and their Viscosity The Corrosion of Iron and Steel The Passivity of Metals Optical Rotary Power The Hardening of Metals The Transformation of Pure Iron Methods and Appliances for the Attainment of High Temperatures in a Refractory Materials Training and Work of the Chemical Engineer Osmotic Pressure Pyrometers and Pyrometry The Setting of Cements and Plasters Electrical Furnaces Co-ordination of Scientific Publication The Occlusion of Gases by Metals The Present Position of the Theory of Ionization The Examination of Materials by X-Rays The Microscope : Its Design, Construction and Applications Basic Slags : Their Production and Utilization in Agriculture Physics and Chemistry of Colloids Electrodeposition and Electroplating Capillarity The Failure of Metals under Internal and Prolonged Stress Physico-Chemical Problems Relating to the Soil Catalysis with special reference to Newer Theories of Chemical Action Some Properties of Powders with special reference to Grading by The Generation and Utilization of Cold Alloys Resistant to Corrosion The Physical Chemistry of the Photographic Process The Electronic Theory of Valency Electrode Reactions and Equilibria Atmospheric Corrosion.First Report Investigation on Oppau Ammonium Sulphate-Nitrate Fluxes and Slags in Metal Melting and Working Physical and Physico-Chemical Problems relating to Textile Fibres The Physical Chemistry of Igneous Rock Formation Base Exchange in Soils The Physical Chemistry of Steel-Making Processes Photochemical Reactions in Liquids and Gases Explosive Reactions in Gaseous Media Physical Phenomena at Interfaces, with special reference to Molecular Atmospheric Corrosion. Second Report The Theory of Strong Electrolytes Cohesion and Related Problems Laboratory Elutriation Orientation Volume Trans. 3 3 6 7 8 9 9 9 10 10 11 12 12 13 13 13 14 14 14 14 15 15 16 16 16 16 17 17 17 17 18 18 19 19 19 19 19 20 20 20 20 20 21 21 22 22 23 23 24GENERAL DISCUSSIONS OF THE FARADAY SOCIETY Date 1928 1929 1929 1929 1930 1930 1931 1932 1932 1933 1933 1934 1934 1935 1935 1936 1936 1937 1937 1938 1938 1939 1939 1940 1941 1941 1942 I943 1944 1945 1945 1946 1946 1947 1947 1947 I 947 1945 1948 1949 1949 1949 1950 1950 1950 1950 1951 1951 1952 1952 1952 1953 1953 1954 1954 Subject Homogeneous Catalysis Crystal Structure and Chemical Constitution Atmospheric Corrosion of Metals.Third Report Molecular Spectra and Molecular Structure Optical Rotatory Power Colloid Science Applied to Biology Photochemical Processes The Adsorption of Gases by Solids The Colloid Aspects of Textile Materials Liquid Crystals and Anisotropic Melts Free Radicals Dipole Moments Colloidal Electrolytes The Structure of Metallic Coatings, Films and Surfaces The Phenomena of Polymerization and Condensation Disperse Systems in Gases : Dust, Smoke and Fog Structure and Molecular Forces in (a) Pure Liquids, and (b) Solutions The Properties and Functions of Membranes, Natural and Artificial Reaction Kinetics Chemical Reactions Involving Solids Luminescence Hydrocarbon Chemistry The Electrical Double Layer (owing to the outbreak of war the meeting The Hydrogen Bond The Oil-Water Interface The Mechanism and Chemical Kinetics of Organic Reactions in Liquid The Structure and Reactions of Rubber Modes of Drug Action Molecular Weight and Molecular Weight Distribution in High Polymers.(Joint Meeting with the Plastics Group, Society of Chemical Industry) The Application of Infra-red Spectra to Chemical Problems Oxidation Dielectrics Swelling and Shrinking Electrode Processes The Labile Molecule Surface Chemistry.was abandoned, but the papers were printed in the T'unsactions) Systems (Jointly with the SociCt6 de Chimie Physique at Volume 24 25 25 25 26 26 27 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 35 36 37 37 38 39 40 41 42 42 A 42 B Disc. 1 2 Bordeaux.) Published by Butterworths Scientific Publications, Ltd. Colloidal Electrolytes and Solutions The Interaction of Water and Porous Materials Trans. 43 Disc. 3 4 Lipo-Proteins 6 Heterogeneous Catalysis 8 Physico-chemical Properties and Behaviour of Nuclear Acids Spectroscopy and Molecular Structure and Optical Methods of In- vestigating Cell Structure Disc. 9 Electrical Double Layer Trans.47 Disc. 10 Hydrocarbons 11 The Physical Chemistry of Proteins 13 The Reactivity of Free Radicals 14 The Equilibrium Properties of Solutions of Non-Electrolytes 15 The Physical Chemistry of Dyeing and Tanning 16 The Study of Fast Reactions 17 Coagulation and Flocculation 18 The Physical Chemistry of Process Metallurgy Crystal Growth 5 Chromatographic Analysis 7 Trans. 46 The Size and Shape Factor in Colloidal Systems Radiation Chemistry 12GENERAL DISCUSSIONS OF THE FARADAY SOCIETY Date Subject 1955 Microwave and Radio-Frequency Spectroscopy 1955 Physical Chemistry of Enzymes 1956 Membrane Phenomena 1956 Physical Chemistry of Processes at High Pressures 1957 Molecular Mechanism of Rate Processes in Solids 1957 Interactions in Ionic Solutions 1958 Configurations and Interactions of Macromolecules and Liquid Crystals 1958 Ions of the Transition Elements 1959 Energy Transfer with special reference to Biological Systems 1959 Crystal Imperfections and the Chemical Reactivity of Solids 1960 Oxidation-Reduction Reactions in Ionizing Solvents 1960 The Physical Chemistry of Aerosols 1961 Radiation Effects in Inorganic Solids 1962 The Structure and Properties of Ionic Melts 1963 Inelastic Collisions of Atoms and Simple Molecules For current availability of Discussion volumes, see back cover. Volume 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

ISSN:0366-9033    

DOI:10.1039/DF962330X001

出版商:RSC    

年代:1962

数据来源: RSC

摘要:

I. COLLISIONS PROCESSES NOT INVOLVING CHEMICAL REACTIONS Resonance Effects in Atom-atom Collisions BY D. R. BATES Dept. of Applied Mathematics, Queen’s University, Belfast, Northern Ireland Received 1st February, 1962 Neglecting the diagonal matrix elements and making other simplifications, an approximate formula is obtained for the cross-section of an excitation or charge-transfer collision. This formula is evaluated in some representative cases. The resonance peak is found to be very sharp at thermal energies. However, the effect of the diagonal matrix elements is likely to be important. In some cases it increases and in others it decreases the sharpness of the resonance peak. This paper is concerned with the influence of the closeness of the energy balance on the cross section for atom-atom collisions involving excitation transfer A+ B*-+ A* + B, (1) or charge transfer A+B++A++B.If the energy balance is exact the process is said to be in resonance. The resonance is called symmetrical if A and B are the same species of atom and asymmetrical (or accidental) if A and B are different species. In the calculations to be described the nuclei are treated as classical particles. States other than the initial state p and the final state q are ignored. Atomic units are used except where otherwise specified. GENERAL THEORY Neglecting the effect of the inter-atomic force on the relative motion, let one of the nuclei be located at the origin of a fixed co-ordinate system and let the other travel with constant speed t, parallel to and at a distance p from the Z-axis.On the two- state approximation adopted expansion coefficients cp(Z,p) and cq(Z,p) associated with the states p and q occupied by the atomic systems on the left and right of (1) or (2) satisfy the coupled differential equations, and the Y’s (which are functions of 2 and p) being the indicated matrix elements of the interaction potential and the E’S the indicated internal energies of the separated systems.l 78 RESONANCE EFFECTS I N ATOM-ATOM COLLISIONS The boundary conditions corresponding to the colliding systems being initially in state p are The probability that the systems are Snally in state q is I C p ( - %P) I = 1 9 cq(- cv) = 0. (6) P@) = I Cq(mYP) f (7) and the cross-section for process (1) or (2) is f* NEGLECT OF DIAGONAL MATRIX ELEMENTS If the difference * between the diagonal matrix elements Vpp and V& is neglected (3) and (4) reduce to and iac, = Tc v exp (-- iaz).az v qp Taking V& and Vgp to be real and equal, and putting VPqdZ = a@) Km and V, exp (iaZ)dZ = A(p9v), s:* Rosen and Zener 2 have conjectured that the required solution to these equations is such that the transfer probability A2 o2 9 = - sin2 { +}. The conjecture is true if the interaction is weak enough for the first Born approxi- mation to be valid or has a form of a certain class (e.g.¶ case (iii) below). It is also true if the energy balance is exact; and recent calculations by Skinner 3 show that it provides at least a fair approximation if the cancellation within the integral of (1 3) is not so severe as to make A smaller than S2 by a large factor.? For present purposes (13) is adequate.To evaluate the transfer cross section from (8) in the low velocity region defined by 1 1) 2’ -Q(O) > It * This difference vanishes in the case of symmetrical resonance but does not in general do so in the case of asymmetrical resonance. t When the cancellation is severe, 9 decreases veryrapidly as a is increased. Instead of being regarded as involving a considerable displacement along the 9-axis the error in the Rosen-Zener approximation to the 9 against a curve may be regarded as involving only an unimportant displace- ment along the a-axis.D. R . BATES 9 it is sufficient to use a slight generalization of a remarkably accurate simple approxi- mation due to Firsov.4 According to this where 3 = A2/2Q2 andp* is the greatest root of the equation 1 1 -Q(p*) = -.V n Using (15) the expressions given below were obtained for the transfer cross- sections associated with three spherically symmetrical, but otherwise representative, forms of the transition matrix element. As usual &(x) is the nth order modified Bessel function of the third kind. CASE (i) (EXCITATION TRANSFER, DIPOLE TRANSITIONS) If the transition matrix element V,, = A/(a2 + p2 + Z2)>5 the transfer cross-section is Q, = na2E*p(a2 +p2)Kf[a(aZ +p2)*]dp (19) 2 A / ~ ( a ~ + p * ~ ) = 1/n. (20) with At resonance (19) becomes Qi = n[,--i;]; nA a2 ' a = 0; while if the energy balance is not close it yields that (22) 32 25 1 Qi-G[(aa)2+-Jaa)+-+ n2 7 . . . exp (-2aa); a a a l ; p*%a.CASE (ii) (EXCITATION TRANSFER, QUADRUPOLE TRANSITIONS) If Vpq = B/(b2+p2+Z2)0, the transfer cross-section is with 4B/3v(b2 + p*2)2 = l/n. At resonance (24) becomes Qii=n[-$} * -.]: b2 a = 0 ; while if the energy balance is not close it yields that (27) 1 Q i i - ~ [ ( b u ) 4 + $ b u ) 3 + ~ ( b a ) 2 + 1 2 g ( b a ) + n2 23 2079 . . . exp (-2ba); ba$l; p * $ h10 RESONANCE EFFECTS IN ATOM-ATOM COLLISIONS CASE (iii) (EXCITATION TRANSFER, AN s--s OR SPIN-CHANGE TRANSITION; CHARGE TRANSFER) If VP, = (++ CC exp [Y{1- (c2 + p2)+ }Isech [(c27;2)*]9 the transfer cross-section is with and with At resonance (27) becomes (nC/yv) exp [y(1- (c" +p*")+/c)] = l/n, (30) x1 = na/2y, x2 = na(c2 +p*2)+/2yc. (3 1) while if the energy balance is not close it yields As already mentioned the Rosen-Zener formula is here exact.Stueckelberg 5 has carried through the very complicated mathematical analysis which arises in the wave treatment. Part of his investigation was, like the present, concerned with transfer in the absence of a pseudo-crossing of the initial and final potential energy surfaces. Account was not taken of matrix elements other than the transition matrix element V& which was assumed for simplicity to be spherically symmetrical and to be of the inverse power law type. The formulae displayed above are in quite satisfactory accord with the corres- ponding formulae of Stuckelberg if the energy balance is not close. Moreover, they correcly predict that the transfer cross-section is greatest at resonance, whereas the formulae of Stueckelberg wrongly predict that the transfer cross-section is zero at resonance unless Vpq falls off at least as slowly as the inverse cube of the internuclear distance .Fig. 1, 2 and 3 show computed Q against A& curves at selected values of v (ex- pressed for convenience as the corresponding kinetic energy E of a system of mass 10 on the chemical scale) the parameters occurring in expressions (18), (23) and (28) for Vpg being assigned the following values : a = b =2, c = y = 1, (34) (35) 1 A = 3, 1 or 2, B = 2,4 or 8, c = l 1 169 5 Or (which make the maximum of V& in each of the three cases 1.7 eV, 3.4 eV or 6.8 eV).D. R. BATES 11 The cross-sections at resonance are very large in case (i), and are quite large even in cases (ii) and (iii).They fall off with increasing rapidity as I A& I is increased from zero. As would be expected the fall off is especially sharp if Eis low or if the range -7 E = lOeV A I I A I 1 I I 10 16' L3 Id ' 16' lCj2 10' lo3 I 2 change in internal energy l d ~ l in electron volts FIG. 1.-Q against A& curves for case (i) with parameters as in (34) and (35) of text. Note that the horizontal scale is broken and that the vertical axes on the left (each with three points marked) change in internal energy ld&I in electron volts FIG. 2.-Q against A& curves for case (ii) with parameters as in (34) and (35) of text. Note that the horizontal scale is broken and that the vertical axes on the left (each with three points marked) are for exact energy balance.of Vpp is long. In the region away from the resonance peak the cross-sections are insensitive to A, B and C, that is, are insensitive to the magnitude of a transition matrix element of given form. Examination of (15) provides the explanation : thus the dependence of Q on A, B or C arises only thronghp" ; and in the region concerned12 RESONANCE EFFECTS IN ATOM-ATOM COLLISIONS the oscillatory nature of the integrand on the left of (12) makes is as great as this upper limit. negligible before p INCLUSION OF DIAGONAL MATRIX ELEMENTS As already noted Vpp and Vqq are equal in the case of symmetrical resonance so that they disappear from eqn. (3) and (4) owing to cancellation. It does not follow that Q is independent of these diagonal matrix elements since (3) and (4) ignore the change in the relative motion arising from them. The effect which this change has on Q is likely to be appreciable if the value of the diagonal matrix elements at inter- nuclear distance p* is comparable with the energy of relative motion.Buckingham and Dalgarno 6 have shown that excitation transfer between metastable and normal helium atoms is greatly inhibited at energies below about 0-3 eV. FIG. 3.-Q against A& curves for case (iii) with parameters as in (34) and (35) of text. Note that the horizontal scale is broken and that the vertical axes on the left (each with three points marked) are for exact energy balance. If the transfer is not of the symmetrical resonance type the change in the relative motion is, in general, much less important than the presence of the diagonal matrix elements in the exponents on the right of (3) and (4).Consider the conditions for the two effects to be negligible. They are respectively, where E is the energy of relative motion, M is the reduced mass and R, is an ill-defined effective collision radius. Expressing Vpp, VQQ and E in eV and ex- pressing M on the chemical scale, but keeping Rc in atomic units, (37) may be written It may be seen that (38) is a more severe condition than (36) unless Vpp and V& are almost equal. If the energy balance is not close, the transfer cross-section determined from the simplified equations, (9) and (lo), is moderate or small at thermal energies (cf. fig. 1, 2 and 3). The effective collision radius R, cannot be large and condition (38) is unlikely to be satisfied.Eqn. (3) and (4) should therefore be used. If fVpp- Vqg] and [ E ~ - E ~ ] have the same sign it is apparent that these equations tend to yield lower 1 Vpp(Rc) - bq(R3 1 / E .e 1mM+@l* (33)D. R. BATES 13 cross-sections than do (9) and (10) ; that is, the distortion due to the diagonal matrix elements tends to increase the sharpness of the resonance peak. The position is different if [Vpp- V&] and [ E ~ - E ~ ] have opposite signs. In this circumstance the tendency is for the distortion to make the resonance peak less sharp (irrespective of whether or not there is a pseudo-crossing of the initial and final potential-energy surfaces). The rather complicated effect of the diagonal matrix elements should be borne in mind in any attempt at arranging measured transfer cross-sections into a pattern showing the influence of the closeness of the energy balance. This work was supported by the U.S. Office of Naval Research under Contract N 62558-2637. 1 cf. Bates, Quantum Theory I Elements (Academic Press, New York, 1961), chap. 8. 2 Rosen and Zener, Physic. Rev., 1932,40, 502. 3 Skinner, Proc. Physic. SOC., 1961, 77, 551. 4 Firsov, J. Expt. Theor. Physics (U.S.S.R.), 1951,21, 1001. 5 Stueckelberg, HeZv. phys. Acta, 1932, 5, 370 ; cf. Mott and Massey, Theory of Atomic Col- 6 Buckingham and Dalgarno, Proc. Roy. SOC. A, 1952, 213, 506. lisions (Oxford University Press, London, 2nd ed., 1949), chap. 12.

ISSN:0366-9033    

DOI:10.1039/DF9623300007

出版商:RSC    

年代:1962

数据来源: RSC

摘要:

Resonance and Non-Resonance Intermolecular Energy Exchange in Molecular Collisions BY E. E. NIKITIN Institute of Chemical Physics, Vorobyevskoye chaussie 2-b, MOSCOW V-334, U.S.S.R. Received 17th January, 1962 Two approximations involved in the calculation of the probabilities of transition between mole- cular vibrational or electronic terms are discussed. An exact solution for two terms is found. In the constant velocity approximation, when the effect of the turning point is neglected, this solution may be used in calculating the charge exchange cross-section in accidental resonance or intramolecular energy exchange in the collision-induced Fermi resonance in three-atomic molecules. The prob- ability of transition between non-parallel vibrational terms is calculated in the semi-classical ap- proximation, the effect of the turning point being taken into account.This is of importance for inelastic molecular collisions involving two or more vibrational modes. The general connection between the Landau-Zener and Landau-Teller approximations is considered and the validity of the classical relative motion of molecules is discussed. Modern theories of inelastic atomic and molecular collisions resulting in electronic or vibrational excitation are based largely on calculations of the probability of non- adiabatic transition between electronic or vibrational terms. In many cases these processes induced by non-adiabatic interaction of intra- and intermolecular motions, can be described in terms of the semi-classical approximation of relative translational motion even for the transition close to the turning point.1 But the more simple cases are those when the main contribution to the transition probability is from the regions that are far enough from the turning point.One example is the crossing of zero-order terms far from the turning point; then the non-adiabatic transition probability can be calculated according to the Landau-Zener formula 2, 3 corrected for the high energy limit by Bates4 and Mordvinov and Firsov.5 In deriving the formula it is assumed that the perturbation matrix element e2 is time independent and the ratio of 82 to the term splitting at infinity is diminishingly small. However, in many cases the asymptotic splitting is not sufficiently large, as in the accidental resonance-type reaction discussed by Bates.6 On the other hand, vibrational terms in non-elastic molecular collisions run almost parallel for all intermolecular distances, and the corresponding transi- tion probability is given by the Landau-Teller formula7 derived for the three- dimensional system by Schwartz and Herzfeld.8 The cross-sections for non- elastic molecular collisions differ very markedly in these cases.In particular, crossing of electronic-vibrational terms in collisions of two NO molecules results in a con- siderable increase (by 3-4 orders) of the probability of vibrational transition from the ground to the first excited vibrational level, as compared to the respective transi- tion probability for 0 2 or N2.9. 10 In this connection it is of interest to find an exact solution of a non-stationary two-state wave equation that might be used for inter- preting all the intermediate cases-from crossing to parallel running terms.Further, we shall consider non-adiabatic transitions in the Fermi resonance induced by molecular collisions. 14E. E. NIKITIN 15 Such processes that are due to molecular collisions are very important for spectro- scopy and kinetics. Besides, solution of this problem not only gives the transition probability for all closely spaced vibrational terms in complex molecules, but also determines the zero-order wave function that can be used in calculating the vibra- tional relaxation time in gases composed of polyatomic molecules. The Fermi-resonance type interaction in complex molecules is known to be one of the main effects that give rise to strong interaction of normal modes.Under stationary conditions the interaction is manifested in abnormal shifts of vibrational terms that are in resonance, and in the mixing of corresponding wave functions. Under non-stationary conditions, when vibrational terms approach each other adiabatically becoming resonant under the influence of external force, the induced Fermi resonance is expected to bring about considerable intramolecular intermode energy exchange. Let us consider a linear three-atomic molecule in its completely symmetrical vibrational states. Now, we suppose that the energy difference 81 between the stretching mode quantum hcol and the doubled bending mode quantum 2ha2 is small as compared with ha1 (here and hereafter h is the Planck constant divided by 2n).In this case the anharmonicity mixes to a considerable amount only functions of the two zero-order states corresponding to energies of hwl and 2ho2; let these functions be 41 and 4 2 . When an external force action is adiabatic with respect to transition between all zero-order vibrational levels, we may restrict ourselves to two function bases. When TO is the characteristic time of force action these conditions may be written as follows : For molecular collisions TO may be taken as p/u, where p is the range of action of the intermolecular potential, and 0 is the relative velocity of molecules. &2/h, 1/20 <mi. 1. THE CONSTANT VELOCITY APPROXIMATION The non-adiabatic wave function may be written as Y = al+l exp ( -$E1dt)+a,42 exp ( - i P 2 d t ) , where El and E2 are zero-order energies without anharmonicity; equations hia, = E~ exp [kbEdf]a2, hia, = E~ exp - - AEdt a,, AE = [ b s 1 When the minimum adiabatic splitting of exact terms (i.e., exact with respect to two function bases) is much smaller than the asymptotic zero-order term splitting, we may put E = y t and ~2 = const.Solution of (3) under these conditions leads to the Landau-Zener formula. However, in our case at least on one side of the crossing, the splitting A23 is not great as compared with ~2 and the Landau-Zener approximation is not valid. We assume that the condition A E % E ~ is fulfilled near the turning point of relative motion of colliding molecules, and approximate the splitting AE by the function AE(t) = ~~a exp [ - F o ~ / E , ] + E,. (4) This corresponds qualitatively to an exponentially decaying intermolecular ex- change interaction and contains two essential parameters-the term splitting at infinity ~1 and the force difference AF at the crossing point.The value a must not16 INTERMOLECULAR ENERGY EXCHANGE enter the final expression for transition probability as long as it is connected with the time reference scale. Here we put v to be a constant because the turning point is far from the crossing. As to the time dependence of Q, it can safely be neglected. This is due to the fact that the change in the anharmonic constant in molecular collisions is small. Introducing new variables z = ~ 2 t / h , a = AFvh/~1~2, p = 4 ~ 2 , (3) can be written in the form s da12 - 1 2 - exp [+i f(~)dz]a,,~; f(z) = a exp (-az)+p, dz where the plus and minus signs in the exponent correspond to the first and second subscripts in a.After replacing c12 = a12 exp [&(i/2)Jf(z)dz] this system reduces to one equation, One of its solutions is cl(z) = A exp (-z/2)z'"*~(is1, 2is0 + 1,z) (7) where z = - (ai/a) exp (- az), s = (1 + p2/4)*/a, s1 = -p/2a+s0, and is the conftuent hypergeometric function. Other solutions of (6) can be ob- tained from (7) by replacing 0 by another linearly independent solution of the cor- responding confluent hypergeometric equations. The asymptotic wave function (2) at z+ + 00 is As follows from (3), 41 and 4 2 are stationary wave functions at z+--co, but at z+ + 00 the stationary wave functions are linear combinations of 41 and 4 2 .Apparently in the limit of small interaction, = 41+m142 correlates with $1 and To calculate the probability of non-adiabatic transition between two stationary states from z = + 00 to z = - 00 (e.g., $I+) we need solve (6) under the boundary condition (8) with p+ = 0. Eqn. (7) is seen to be such a solution. The factor A must be found to satisfy the asymptotic behaviour c1(z) at z+O (z- + GO), $2 = as141 - 4 2 with 4 2 . From the condition of crossing we have a<O and Im(z)>O and thereby On the other hand, the following asymptotic expansion is valid for 1 z 19 1,11 A = exp ($nso)/(l + a2si)+. (10) where s1 +s2 = Bo. Thus, the transition probability ~($1-41) = p11 is given by and from the normalization condition it follows that p($1-)42) = 1 -p11.E.E. NIKITIN 17 To obtain the transition probability of the reverse process (z = - co -+z = + co), This choice we must use in (7) a solution of (8) that would vanish at T = -a. is accounted for by the boundary conditions of the transition, I c , I +O, l j c , I 3 1 at 2 3 - co. (1 3) An appropriate solution is (1 1) exp (z)Y (1 + is2, 1 + 2is0, - z), and the corres- ponding expansion is r( - 2isJ 1x2 is,) r(1- is,) r(l +is, +2is,) Y = + (-z)-2iso, I z I Q 1. From (13) we have Thus, comparing the behaviour of c1 at I z 14 1 in (14) with that of (8) corresponding to we obtain the following expression, Allowing for the identity 1 + a2sf = 2a2s0s17 (12) and (17) may be rewritten in the simple form These equations can be shown to be valid not only for crossing terms (a < 0, p> 0), but for non-crossing zero terms as well.In the latter case we must formally put p c 0 to reverse the terms at z+ + 03. To obtain the transition probability 9 1 2 for double passing of the region of great non-adiabacity we can use the relation, Here, however, 9’12 is the transition probability averaged over a small range of velocity changes. After such an averaging, all the terms that are due to interference between a1 and a2 in the second passage vanish. Thus, from (18) and (19) we obtain 9 1 2 = 2P12P22. (19) sinh2 (2ns0) This expression takes the simplest forms under the following conditions. Then, and (20) becomes the Landau-Zener formula, (i) Great asymptotic term splitting and small velocity : &1/&2% 1, 2n$/AFhv% 1.2ns, -+2x~z/AFhv; s2, so-+ CQ P12 = 2 exp (-2n~;/AFhv)[l- exp (-2n$/AFhv)].18 INTERMOLECULAR ENERGY EXCHANGE (ii) Exact resonance to the zero approximation: EI = 0. Then, introducing for convenience a new parameter-the effective range of action of intermolecular forces p = &l/AF-we obtain 9'' = +~h-' (m2p/hv). This result coincides with the formula devised by Rosen and Zener,l2 if the oscil- lating factor in the latter is replaced by 3. (iii) The great velocity of molecules : 274, 27rne:-g AFhv. After expansion (20) becomes The maximum value of 9 1 2 in all cases is 9 1 2 = 3. 2. THE CLASSICAL MOLECULAR MOTION APPROXIMATION In the above section we assumed the condition that the turning point is suf- ficiently far from the region making the main contribution to the non-adiabatic transition probability.This is valid if the velocity change Av during the time of strong interaction At is small as compared to u. Thus, e.g., for crossing terms we may write where m is the effective mass of colliding molecules. Then the above-mentioned condition will be A P v ( A ~ ) ~ / ~ ~ - 1 and Av - (F/m)At, (24) mv E()f<l. AFv In thermal molecular collisions this condition is in many cases equivalent to the low value of intermolecular energy at the crossing point (or at the region of essential term divergence if the terms do not cross) to the mean thermal energy kT. How- ever, in cases when the change in splitting A E is relatively small and its asymptotic value is sufficiently large, the main contribution to the transition probability can also be that from the region near the turning point.Then the constant velocity approximation is not valid and all the equations of $ 1 fail at the limit a+O. But this case is of practical importance because it takes place in molecular collisions resulting in translational-vibrational energy exchange involving only one vibrational mode. Fortunately, for small a, calculation of 9 1 2 can be carried out by the perturbation method. To follow the transition from constant velocity approximation to the more accurate classical molecular motion approximation that takes into account the velocity change of colliding molecules close to the turning point, let us take as a zero-order basis the functions $1 and $2. Then, splitting AE' of these zero-order terms (in 82 units) and the perturbation matrix element V12 are AE' = 2 sin-' #+a exp(-"') cos #, V12 = $a exp(-"') sin #, (26) where cotan 4 = 3p.When 9 5 2 < 1, the exact expression (20) can be obtained ap- proximately by using the perturbation method. For instance, at j?+ - co(4-2//3), the asymptotic expansion of (20) givesE . E. NIKITIN The same result can be obtained from the equation, 19 P12 = 2 ~ ~ ~ m V , , e x p ( i ~ ’ d r ) d z ~ ’ = The last expression does not depend on a because the integrand is invariant (with an accuracy to the phase factor) when replacing z+(z+const.). But this is not a case for classical motion approximation. Allowing for the effect of the turning point we have to replace function a exp (- az) in (28) by another function $(z) that accounts for the true molecular motion in collision.In our approximation the intermolecular interaction vanishes exponentially with the molecular separation, so that we have X(z)-a exp (- a 1 z 1). On the other hand, at small z that corres- I = I-+a pond to close molecular approach, ~ ( z ) should have a maximum corresponding to the turning point. Thus, the allowing for the turning point leads to the expression, As an example we can take ~ ( z ) = (a/2) cosh-1 az. perturbation method using terms in (26) with p = 0 gives Calculation according to the g12 = (7~a/4.)~ cosh-’ (7~12). gI2 = sin2 (za/4a) cosh-’ (n/a). (30) (31) The exact solution of the corresponding equation found by Rosen and Zener 12 is Thus, it may be seen how (30) and (22) approximate this exact solution for 9’12.For a real case, the function ~ ( z ) must be calculated as a perturbation matrix element, the time dependence of which is determined by the classical motion of elastic-scattered molecules. Taking the most common model used in the theory of vibrational-relaxation of molecules, we assume that the exponential repelling intermolecular potential is of a characteristic length p and the perturbation is linear with respect to the vibra- tional co-ordinate x. Then ~ ( z ) will be ~ ( z ) = (a/4) coshe2 (taz). Calculation according to (29) gives Using the asymptotic expansion and introducing (34) and (33), we obtain, after averaging the rapid oscillating cos2 $(&a), either the Landau-Zener expression (if /?a <O), or (27) (if aa> 0).On the other hand, using (33), we can consider the limiting case of small a, when the last factor may be dropped.20 INTBRMOLECULAR ENERGY EXCHANGE The coefficients a in the exponentials of (25) and (32) and the pre-exponential can be conceived as having a different meaning, as we know beforehand that the transition probability must be proportional to the squared matrix element of per- turbation. For the above-mentioned model we put where x12 is the transition matrix element of the vibrational co-ordinate, AEI the additional term splitting at the turning point with respect to asymptotic splitting 81. The general computation of (33) at any values of parameters /3 and a is not difficult for pa> 0, because in this case @ reduces to the hydrogenic s-wave function for the wave number k = a/l p I and R = ap/2a2, values of which are tabulated.But in many cases (e.g., for vibrational excitation) the parameter p/a is sufficiently large, so that the last factor in (33) can be put in asymptotic form, (36) I is bounded. is large, r = - = a hu a2 h2v2 B E1P -=- Apparently for A E ~ / E ~ Q 1, we can put A E ~ -rnv2/2, so that velocity, and the averaging procedure does not affect this factor. does not depend on 3. AVERAGING OVER THERMAL DISTRIBUTION Averaging of 9 1 2 over the impact parameter for the spherical-symmetrical inter- action is not difficult, in principle, and can be performed for most of the cases using, for instance, the “modified wave-number method” set out by Schwartz and Herzfeld.8 The averaging over velocities is more complicated because on the classical description of relative molecular motion the effect of inelastic transition on this motion is neglected.This means that for non-crossing terms the kinetic energy of molecules at infinity is defined with an uncertainty 81. If the latter is neglected, the averaging gives though the principle of detailed balancing requires When kT, the difference is not essential, because (37) and (38) coincide to the zero order. Then (38) can be fulfilled if 9 1 2 and 9 2 1 are modified to the first order that does not give any appreciable change in probabilities as such. This is a usual pro- cedure for the derivation of a semi-classical relaxation equation,13 but when el + kT all the above functions must be symmetrized before averaging.14 The reason for this is as follows.We know that in exact quantum-mechanical calculation the transition probability must be a symmetrical function of initial VQ and final vf veloc- ities of molecules. In the limiting case of quasi-classical motion we must obtain the above equations, where v is sume symmetrical function of vf and uf. But for any symmetrical function we can write (912) = (912) (37) <912>/<921> = exp (&llkT) (38) v(vi,vf) = virfrsl/2mvi+ . . . (39) where the plus (minus) sign corresponds to deactivation (activation). The first two terms of this expansion with respect to the small parameter sl/mv,2 do not depend on the kind of the function ~ ( u Q Y ~ ) , while the others do. The latter conhe ourE. E. NIKITIN 21 application of classical motion approximation to calculation of transition probab- ilities.The classic motion approximation is valid when the transition probabilities are not appreciably affected by the relative uncertainty in molecular energy or velocity of the order (~/mv:)2. This is the order of terms neglected in (39). The introduction of (39) in (33) gives the transition probability that can be de- rived using the method of distorted waves and it has been used by Schwartz and Herzfeld 8 to obtain a quantum-mechanical version of the Landau-Teller formula. It may be seen that classical motion approximation introduces an uncertainty factor q-exp rr:ip - ( - m::*)2)2] in the Landau-Teller formula. Now v* is velocity that represents the main contribution to the average probability. Averaging over velocities by the steepest descent method gives- = x* = (n2mp2&:/2h2kT) and q-exp [i$/(kT)2x*]. Almost in all cases, q is close to unity and the corresponding correction is usually neglected even in the quantum-mechanical calculations. Upon averaging, (33) can be used for calculations of the vibrational transition probabilities in many-atomic molecules for which the non-parallel vibrational terms are very probably due to participation of different vibrational modes. m(v*)2 2kT The author is greatly indebted to Prof. N. D. Sokolov for helpful discussions. 1 Nikitin, Optika Spektr. Russ., 1961, 12, 452. 2 Landau and Lifshitz, Quantum Mechanics (Gostekhizdat, Moscow, 1948). 3 Zener, Proc. Roy. SOC. A , 1932, 137, 696. 4 Bates, Proc. Roy. SOC. A, 1960, 257, 22. 5 Mordvinov and Firsov, Zhur. Exp. Theor. Phys. Russ., 1960, 39, 437. GBates, Proc. Roy. SOC. A , 1959, 253, 141. 7 Landau and Teller, Physik. 2. Sowiet, 1936, 10, 34. 8 Schwartz and Herzfeld, J. Chem. Physics, 1954, 22, 767. 9 Nikitin, 8th Int. Symp. Combustion (Waverly Press, 1961). 10 Nikitin, Optika Spektr. Russ., 1960, 9, 16. 11 ErdClyi, Magnus, Oberhettinger and Tricomi, Higher Transcendental Functions (McGraw Hill, 12 Rosen and Zener, Physic. Rev., 1932, 40, 502. 13 Hubbard, Rev. Mod. Physics, 1961,33,249. 14 Nikitin, Optika Spektr. RUSS., 1959, 6, 141. 1953), vol. 1.

ISSN:0366-9033    

DOI:10.1039/DF9623300014

出版商:RSC    

年代:1962

数据来源: RSC

摘要:

Deactivation of Vibration by Collision in CO, BY KARL F. HERZFELD* Dept. of Physics, Catholic University of America, Washington 17, D.C. Received 16th January, 1962 COZ has one degenerate bending vibration, one totally symmetric and one asymmetric valence- bond vibration. There are a number of different ways by which these can be deactivated by collision. The rates are calculated in the range from 300 to 2500°K and compared with experiments. There seem to exist a contradiction between the results of different experiments. 1. THE PROBLEM The carbon dioxide molecule has three different characteristic modes of vibra- tion : 1 (i) the bending vibration, doubly degenerate, which has a characteristic temperature 8 = Rv/k = 960°K. This is conventionally called v2; (ii) the totally symmetric valence bond vibration, in which the C atom remains at rest, with 81 = 1920; (iii) the asymmetric valence bond vibration, in which C vibrates against the two oxygen atoms, with 03 = 3380.We ignore in the following the Fermi resonance, due to the near coincidence of 282 and 81. The problem considered in the following is the deactivation of these vibrations by collisions, as function of the temperature. This deactivation may occur directly, or by complex collisions.2 For convenience, we measure also the kinetic energy in temperatures, so that a kinetic energy increase by 500°K means an increase of the translational energy of a molecule by 500 k. The following is the list of possible deactivation processes, which will be designated subsequently by Roman numbers. v,; e2 = 960 v , ; el = 1920 direct : direct : or 0, = 960-kinetic energy 960; or O1 = 19204920; hv, + kinetic energy ’ hv, + kinetic energy ’ &+& +960 kinetic energy ; 1920+960+960 ; resonance: 81 +202 ; lt920+2 x 960 ; v,; e, = 3380 direct : , 3380-3380; hv, -+ kinetic energy &-’&+kinetic energy ; 3380+960 - 2420 ; O3 +el + kinetic energy ; 3380+ 1920 -I- 1460 ; 03+282 +kinetic energy ; 3380+2 x 960+ 1460 ; O3 +& + O2 + kinetic energy ; 33804 1920 + 960 + 500 ; 03+302 + kinetic energy ; 3380+3 x 960 + 500.of these processes have been calculated in this paper for the temperature range 300 to 2500°K. Recent measurements are now available. Additional interest comes from the hypothesis of Laidler et aZ.,3 that energy transfer does not easily occur between symmetric and antisymmetric vibration.Although it must * aided by Office of Naval Research. 22K. F. HERZFBLD 23 be emphasized that we only consider energy transfer during a collision, it will be interesting to see whether a similar rule is valid here. It would make processes VII and IX impossible.* 2. THE METHOD The methods to calculate the rates are given in the book of Herzfeld and Litovitz 4 and will not be repeated herz. Intermediate data used in the calculation are given in appendix B. However, several remarks must be made. For the rate, we write in accordance with the book 4 rate = (ZZJ , where zc is the time between collisions, defined as q = 1*271p~, 2 is independent of pressure for an ideal gas and is approximately the number of collisions needed for deactivation.Unfortunately, numerical errors have crept into table 66-5 of the book which are corrected here. The formulae contain a steric factor 20. The present author had deduced an expression for that factor,s which he now believes unjustified because, in the deduction, the interaction potential was divided into one part responsible for elastic scattering and one part responsible for inelastic scattering. This is now believed to be wrong; instead, one should use the averaged potential for elastic scattering. If the methods given in the paper 5 are then used, one finds the original value 20 = 3 for direct deactivation of longitudinal vibrations, and 20 = 3/2 for direct deactivation of the doubly degenerate bending vibration. For the complex deactivations, we write 2 0 = 320/3 leaving 2 0 undetermined theoretically.The equations given by Schwartz 2 apply directly to non-degenerate modes of a molecule hit by a different molecule. Witteman6 has pointed out that in the collision between two C02 molecules, and because of the degeneracy of the bending vibration, many more possibilities exist and that, therefore, the rate of a definite complex collision in which the mode which is excited is specified has to be multiplied by a number > 1 to get the total rate, or 2 divided by that number to get the correct 2. For example, in process 111, OpO2+kinetic energy, four modes 0 2 are available for excitation. The details are given in appendix A. Witteman makes definite assumptions about the 20 for each possibility.Instead, an average 20 is left open for each complex collision. 3. RESULTS Table 1 lists the specific heats (in units R) for the different modes. These have been taken without any refinements like Fermi resonance, from the Planck-Einstein formula, using tables in the Mayer-Mayer book? TABLE 1 VALUES OF c/R T, "K = 300 600 1000 1500 2000 2500 92 0.9072 1.6230 1.8514 1 -93 30 1.9618 1.9756 01 0.0688 0.4536 0.7422 0.8742 0.9257 0.9522 0.1 146 0.4168 0.6666 0.7930 0.8609 9 3 * It should be added that other processes than those enumerated above, I to X, are possible at higher temperatures, in which more than a single excited mode is deactivated, e.g., 01+02+2760 kinetic energy. Such processes have been neglected since their probability is very small.4 RELAXATION IN c02 Table 2 shows for the complex collisions those values of 2 which one gets if the mode to be deactivated is in quantum state 1, all the other modes in quantum state zero and only jumps by unity occur.Table 3 shows the calculated 2 for all processes considered here; for the complex collisions, the values of table 2 have been divided by the “ weights ” for which the formulae are given in appendix A. TABLE 2 z FOR A COMPLEX COLLISION IN WHICH THERE IS A TRANSITION 1+0 AND TRANSITIONS O-tl IN DEFINITE MODES T. OK = process I11 IV VI VII VIII IX X 300 1-lox 106 7400 1 . 8 ~ 1012 1 . 6 ~ 1010 1 - 6 ~ 106 1 . 8 ~ 109 2 . 3 ~ 107 600 1000 1500 2000 2500 68,000 9000 2600 1100 420 5600 2900 2100 1600 1360 1 . 4 ~ 106 4 . 6 ~ 107 3 . 7 ~ 106 585,000 180,000 74,000 4 .6 ~ 108 3.5X 107 5 . 7 ~ 106 1.71 x 106 726,000 284,000 92,200 39,200 22,900 15,600 4.3 x 106 1.4 x 106 618,000 383,000 252,000 TABLE 3 EFFECTIVE (TO BE MULTIPLIED BY UNKNOWN zo/3 EXCEPT FOR I, II AND V) T, OK = 300 600 1000 1500 2000 2500 process I (960) I1 (1920) I11 (1920) IV (1920) V (3380) VI (3380) VII (3380) VIII (3380) IX (3380) X (3380) 16,300 1 . 4 9 ~ 108 265,000 490 4-4x 1011 8 . 8 ~ 108 193,000 392,000 4.7 x 1013 1 . o ~ 109 1160 220 95,200 14,200 1600 270 94 2 . 2 ~ 107 1 . 6 ~ 106 2-1 x 107 981,000 27,300 6280 41,900 6670 73 37 4100 430 170 38 28 297,000 66,400 101,000 21,700 1870 810 1400 510 24 1700 43 8-8 1 . 6 ~ 106 104,OOO 26,500 3340 310 310 It is evident from table 3 that the sequence IV (resonance) is most efficient for the deactivation of 01 (provided there is no extraordinarily large ZO), with sequence I11 approaching it in efficiency above 2000°K. For the deactivation of 03, process IX is most efficient (again with the proviso about ZO), but X is comparable over the whole temperature range, so that, if 20 = 3 for both, the combination of processes IX and X gives the results shown in table 4.The Z values for IX and X at 2500”, as given in tables 2, 3 and 4, may be too large, due to the mathematical approximations made. 4. COMPARISON WITH EXPERIMENTS At 300”K, the most direct comparison is with the “spectrophone” in which a particular vibration is excited by absorption of infra-red radiation, and the timeK. F. HERZFELD 25 delay in the pressure increase is measured. Unfortunately, the experiment is very difficult.Slobodskaya,8 who did it first, found 2 = 14,000 for the bending vibra- tion and 63,000 for the asymmetric valence bond vibration. The first agrees very well with the value in table 3, the second is about half the number in table 4. In any case, a large 20 is not compatible with this. Bauer and Jacox’s measurements 9 vary by a factor 2 within themselves (at low pressure). The average 2 for 02 agrees with that of Slobodskaya, but the 2 for 0 3 is equally low. 0 3 cannot be detected i n ultrasonic measurements. Henderson and Klose 10 find (at 323.7”K) a single absorption peak ; however, its height, i.e., the value of 2C2 + C1 calculated from it, is a few percent too high. No explanation of this excess is known. However, 2 for IV cannot be larger than 2 for I.Dr. Wittleman will discuss his unpublished experimental results, which are in fair agreement with those of Winkler and Smiley.11 The latter do not deviate too much from the values of the direct process I according to table 3. The usual shock-tube methods do not work below 2- 300 ; perhaps the procedure of Hornig 12 might be successful in such cases. In the range 2000-250OoK, Greenspan and Blackman 13 give z = 3.5 x 10-6 sec or 2-7000. They interpret this as relaxa- tion of 01 ; this would, however, imply that process IV has a 20 of about 1000 making its room temperature 2 about 160,000 which is in complete disagreement with Henderson and Klose’s result of good adjustment between 01 and 02. Hurle and Gaydon 14 also have made shock-tube experiments around 2500°K.They find relaxation times of 35 and 54 x 10-6 sec, which can be considerably shortened by impurities, particularly water vapour ; so it seems reasonable to assume that Green- span and Blackman’s values have been shortened in this way. Hurle and Gaydon ascribe their relaxation to 03 and think it probable that the asymmetric valence bond vibration cannot easily exchange energy with the symmetric one. This would eliminate processes VII and IX. If we take 45 x 10-6 sec as an average value, one gets 2-77,000, and one must also deny the effectiveness of VIII and X and leave VI as the most probable sequence. This, however, contradicts the spectrophone results at 300°K. We are, therefore, confronted with a contradiction : the spectrophonic experi- ments seem to prove that at room temperature there is no particular difficulty of energy exchange between the asymmetric valence bond vibration and the sym- metric longitudinal vibration plus one bending vibration during a collision (IX and X).The shock-tube experiments seem to prove that, even in a collision, Laidler’s rule holds, namely, that there is no energy exchange between 01 and 03 and in addition that 03 cannot exchange energy with more than one bending mode. APPENDIX A Consider as an extreme example process 0, 03+302+kinetic energy. Let us first assume low temperatures so that all the bending vibrations are in the ground state. Then the following possibilities exist. (a) The three quanta go into different modes. These are 4 possibilities, since any one of the modes may be unaffected.(b) Two quanta go into one mode, the third quantum into a different mode. There are four different choices for the two quantum modes, and for each, three different choices for the additional quantum, or 12 choices; in addition, however, the two quantum jump has a factor 2 ; therefore (b) makes a contribution of 24. (c) All three quanta g o into the same mode, of which there are four. However, the jump from 0 to 3 gives a factor 6, so one has a contribution of 24. The total factor is 4+ 24+ 24 = 52. At higher temperatures, this is further increased because some molecules are already excited, and the corresponding transitions have higher probabilities. For26 RELAXATION IN COz example, if the average state of the oscillators were o = 2, (a) would contribute 4x 3, (b) 12x 12x 3, (c) 4 x 60, or a total factor of 684.Use the expressions C n j = 1, C j n j = e-eJT(l - e-e/T)-l, C j2n = e-e/T(l+ e-eJT)(l - e-'/T)-2, C j 3 n j = e - e / T ( 1 + 4 e - e / T + e - 2 e / T ) ( l -e-@iT)-3, where nj is the fraction of the molecules in state j. Abbreviate Then one has the following " weights ". (111) 4 f d - l (IV) One starts with This leads to 6[CO'+1)nj12+4CO'+1)0'+2)nj. (IV) 14f3,l ; (VI) 4 f 2 P ; (VII) 2flf3l ; (IX) 8flf,fi-1 - (VIII) 14f,2f,-l ; (X) One starts with 4[c(j + l)njI3 + 12[c(j + 1)(j +2)nj][C(j + l ) n j ] + 4x0 + l)(j + 2)(j + 3)np This leads to 52fZfT' . It can be seen that the way in which the different f appear in the " weights " is deter- mined by the type of process, after the fashion of the chemical mass-action law, but I have been unable to give a general proof for this.APPENDIX B It is assumed that the interaction energy between two C02 molecules obeys the Lennard- Jones law 4E[(912-($7. For C02, ro = 3-996A, &/k = 190°K. For the purposes of our calculation, the Lennard- Jones law is replaced by - E + Ho exp (- r/l). The length I is determined so that for the most important kinetic energy of the molecules, Em, the Lennard-Jones function and the exponential function have the same slope. One now defines an energy 8' (measured in units of temperatures) by Eere v is the vibrational frequency, 6 the effective mass in absolute units per noIea.de, M the effective mass per mole (for the collision of two CO2 molecules: this is 22).InK . F. HERZFBLD 27 the last equation I is in A. In a complex collision, 8 is the energy transferred to translational energy. One then has for the energy of the most efficient molecules at temperatute T : Em = +k(O')"T'. The values used in the calculation are shown in table B. T, OK = Ern18 Z in A e'x 10-5 Ern/& ZinA e'x 10-5 Emf& Z in A 8' x 10-5 Ern18 I in A 1~x10-5 Ern/€ IinA e'x 10-5 Ern18 I in A O'X 10-5 300 10-30 0.2005 6-67 1 16.58 27.793 0.2049 24.39 88.580 19.36 44.522 0-2079 0.2059 13.72 0.2030 15-763 6-566 0.1963 1 -727 TABLE B 600 1000 1500 2000 1 AND 111 16.56 23.58 32.55 37.73 0.2047 0.2076 0.2098 0.2113 6.926 7.109 7.240 7.365 I1 37.70 0.2109 29-400 60.19 29.905 0.2127 V VI VII AND VIfl 2207 3 1.23 41-14 49-98 16.375 16.720 16.986 17.130 0.2069 0.2091 0.2108 0.21 17 Ix AND x 10-58 15.04 19.83 24.17 0.2005 0.2041 0.2060 0,2074 1.808 1.868 1 -9025 1.936 2500 44.15 0-2141 7.555 73-32 34.594 0.2135 102.5 0.2146 94.380 81.78 48.02 0-2138 58.13 17.257 0.2124 28-12 0.2087 1.953 1 Herzberg, Infra-red and Raman Spectra (van Nostrand Co., N.Y., 3rd edn., 1945). 2 Schwartz, Slawsky and Herzfeld, J. Chem. Physics, 1952, 20, 1591. 3 Gil and Laidler, Proc. Roy. SOC. A, 1959, 250, 121 ; 1959, 251, 66. 4 Herzfeld and Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic Press, New 5 Herzfeld, 2. Physik, 1959, 156, 265. 6 Witteman, J. Chem. Physics, 1961, 35, 1 . 7 Mayer and Mayer, Statistical Mechanics (Chapman and Hall, London, 1940). 8 Slobodskaya, Izvest. Akad. Nauk S.S.S.R. (ser. Fiz.), 1948, 12, 656. 9 Bauer and Jacox, J. Physic. Chem., 1957, 61, 833. York, 1959). 10 Henderson and Klose, J. Acoust. SOC. Amer., 1959, 31,29. 11 Smiley and Winkler, J. Chem. Physics, 1954, 22, 2018. 12 Hansen and Hornig, J Chem. Physics, 1960, 33, 913. 13 Greenspan and Blackman, Bull. Amer. Physic. Soc., 1957,II, 2, RA9. 14 Hurle and Gaydon, Nature, 1959, 184, 1858.

ISSN:0366-9033    

DOI:10.1039/DF9623300022

出版商:RSC    

年代:1962

数据来源: RSC

摘要:

Vibrational Relaxation of Nitric Oxide BY A. B. CALLEAR Dept . of Physical Chemistry, The University, Cambridge Received 29th January, 1962 A method of over-populating the first vibrational level of the ground electronic states of NO was described, by flashing mixtures of NO with He, Kr and N2. Production of vibrationally ex- cited molecules is due either to fluorescence or collisional quenching of NO . A*C+, according to the conditions. The rates of vibrational relaxation were determined by plate photometry. In experi- ments using high nitric-oxide pressures, the observed temperature rise makes possible the estimation of the quantum yield for conversion of electronic energy to vibrational energy. The rate of decay of NO X2II(u = 1) was increased by inclusion of small partial pressures of H20 and CO in the mixtures.Detection of vibrationally excited CO was presented as evidence for the occurrence of the exchange reaction : NO(# = l)+CO(tr = O)+NO(V = O)+CO(u = 1). The logarithms of the probabilities of vibrational exchange for three different reactions were found to be a linear function of energy difference between the vibrational quanta of the colliding molecules. Vibrational relaxation of nitric oxide has been studied by Robben1 in shock- heated gas using ultra-violet absorption spectroscopy to measure the population of molecules in the first vibrationally excited level. He observed an abnormally fast self relaxation, PO-1 being 103-104 according to the temperature. This fast re- laxation was interpreted by means of sticky collisions between NO molecules.Relaxation processes in nitric oxide have also heen studied recently by Bauer, Kneser and Sittig 2 by sound absorption. They also showed that the process, NO(u = l)+NO(V = 0)+2NO(u = 0), is very fast, the observed probability of vibrational relaxation being 3-7 x 10-4 at 20°C. The present spectroscopic study arose out of examination of the principle that it should be possible to produce an appreciable concentration of vibrationally excited diatomic molecules with a flash of ultra-violet light without causing decomposition. Absorption of light in the ultra-violet and visible regions produces electronically excited species which re-ernit the fluorescence spectrum, populating the excited vibrational levels of the ground electronic states according to the probabilities of the various transitions.Nitric oxide was selected for study (a) because of the occurrence of the y bands in the quartz ultra-violet region and (b) because radiation down to 2000 A causes very little decomposition. The first experiments were unsuccessful because of the high intensity of the fluor- escence ; instead of observing the y(0,l) band in absorption it appeared in emission. After improvement of the optics and recognition of the importance of the pressure broadening effect, the phenomenon was demonstrated satisfactorily and was applied to the study of the vibrational relaxation of nitric oxide and the effect of added gases.3 Information was also obtained about conversion of electronic energy to vibrational energy. This study was later extended to the vacuum ultra-violet region to obtain evidence for the occurrence of vibrational exchange processes.4 Spin orbit relaxation occurred once in 16 collisions.28A . B. CALLEAR 29 The same principle was extended to the CS molecule which can be conveniently produced by photo-chemical dissociation of carbon bisulphide.5 Secondary absorp- tion of light by CS overpopulates the vibrational levels of the ground electronic states though in this case the CS are also produced with some vibrational energy. The simultaneous decay of molecules with up to at least seven vibrational quanta can be observed and the system promises to be of value in studying highly vibrationally excited molecules, first because of the stability of CS which persists for several minutes after flashing, and secondly because of the occurrence of the main band system in a convenient wavelength region.One of the main objects of presenting this paper is to bring together some aspects of the vibrational relaxation of nitric oxide which have been or are still being studied in these laboratories. In the discussion the evidence for occurrence of exchange of vibrational quanta between molecules is examined and an attempt is made to correlate the efficiency of vibrational transfer with the difference in energy of the vibrational quanta of the two colliding molecules. EXPERIMENTAL Most of the experimental details have been described.3~4 For the quartz u.-v. region experiments, the reaction vessel and flash lamp were both 50cm long and enclosed in a cylindrical reflector coated with magnesium oxide.The absorption spectra were obtained using a Hilger-Littrow Spectrograph and a single pass through the reaction vessel. It was important to line up the capillary of the spectroscopic flash very carefully so as to obtain the maximum possible useful light output, because the intensity of the fluorescence was very strong with NO pressures of 5 m or less. Quantitative measurements could only be made under conditions where the blackening of the plates due to fluorescence was very small compared to the plate density obtained from the spectroscopic flash. The intense fluorescence would present considerable difficulties if a photoelectric method was used. Spectra were taken on Ilford HP3 plates using standardized development.Absorption spectra of carbon monoxide were obtained using a 1 m vacuum spectro- graph. The reaction vessel for these experiments was 7.5 cm long, placed along the axis of a flash lamp constructed in the form of a helix, the whole being enclosed in a cylindrical reflector coated with magnesium oxide. Spectra were taken on Kodak V. 6002 experi- mental film. To avoid the appearance of CO bands in the spectroscopic flash, it was re- filled after each exposure. Plates were photometered on a Joyce-Loebl double-beam re- cording microphotometer, model E. 12, &. 111. The preparation of the materials has been described.3 RESULTS AND DISCUSSION PRODUCTION OF NO. x2n(u = 1) If nitric oxide is subjected to a high intensity flash of ultra-violet light, the con- centration of molecules with one vibrational quantum in the ground electronic states (NO.X2II(u = 1)) is greatly overpopulated due to absorption of light by the y bands ; the production and decay of the vibrationally excited molecules can be conveniently studied by kinetic spectroscopy because of the occurrence of the y bands in the quartz ultra-violet region. If an ammonia filter is used to cut out absorption below about 2200 A, the effect is still observed strongly and under these conditions it is due entirely to absorption by the y(0,O) band. The maximum concentration produced by flashing 2 mm of NO with 600 mm of N2, flash energy 1600 J, is about 0.1 mm of excited molecules which is about the same concentration of NO . X2II(u = 1) which is present in a full atmosphere of pure nitric oxide at 20°C.Under these conditions, the absolute concentration of NO . X2rI(u = 1) can be estimated by comparing plate densities of the (0,l) band in pure nitric oxide with those in flashed mixtures of nitric oxide and inert gas. This procedure is not exact because the pressure broadening30 VIBRATIONAL RELAXATION OF NITRIC OXIDE of the y bands under these two different types of conditions is not identical. How- ever, except for the determination of the quantumyield, the precision of the absolute values is not an important factor since the decay of NO. XW(u = 1) is first order, the relaxation time being independent of the absolute concentration. The maximum concentrations of NO. X2II(u = 1) produced and the intensity of the y fluorescence are very dependent on the pressure of inert gas.Fig. 1 shows the increase with nitrogen pressure of the y fluorescence and the (0,O) absorption. Under these conditions where absorption is very strong at the line centres, pressure broadening causes a very marked increase in total light absorption by the broadening of thewings of the lines. A similar effect can be observed if the mercuryresonance line at 2537 A is excited with a continuous light source.6 Fig. 2, 3 and 4 are sets of absorption spectra taken during and after the exciting flash. The NO. X2rI(u = 1) reaches a slightly higher concentration in N2 than in krypton and the decay is slightly faster in N2 than in krypton, though this difference was not sufficiently marked to be recorded satisfactorily by plate photometry.The decay is in fact determined almost entirely by the partial pressure of nitric oxide itself. With helium as inert gas, the maximum concentration of vibrationally excited mole- cules is lower than that produced in N2 or krypton because helium causes less pres- sure broadening of the y bands than the other two gases, resulting in a lower light absorption. By photometering plates of the types given on fig. 2, 3 and 4, the con- centration of NO. X2rI(u = 1) can be expressed as a function of time and some re- sults are plotted on fig. 5. The time zero corresponds to the minimum delay of the photocell circuit and firing unit. The exciting flash decayed to one half of its peak intensity in 25 psec and is effectively terminated by 50 pee, so that by plotting the logarithm of the concentration remaining after the flash with time, the relaxation rates can be determined.MECHANISM OF EXCITATION The first step in the vibrational excitation is formation of NO . A%+(u = n). These species either radiate, or are quenched to overpopulate the excited vibrational levels of the ground electronic states. The latter process predominates, except at very low nitric-oxide pressures, since the self- quenching of the fluorescence is very marked, the half pressure being < 1 mm. Vibrationally excited molecules are only observed in the first vibrational level possibly because fast vibrational exchange processes between nitric-oxide molecules rapidly produce a Bolzman vibrational distribution, the temperature being well above that of the other degrees of freedom.NO. A2Z'(v = n)+NO. X211(v = m)+hv, NO. A2E+(v = n)+NO. X211(v = ())-+NO. X2rI(u = p)+NO . X211(u = q), VIBRATION-TRANSLATION RELAXATION OF NO. X2n(U = 1) From the slopes of the log plots of the data of fig. 5, the first order rate constants The rate constant k can be split into two terms, given in table 1 were determined. The temperature is estimated at 23f.3"C. k = k,(NO) + k,(N,), and the best fit with the data of table 1 leads to for NO = 3.6k0.4 xblank abs. no N2 fluor. abs. 120 mm N2 fluor. abs. 240 mm N2 fluor. abs. 360 mm N2 fluor. abs. 480 mm N2 fluor. abs. 600 mm NZ fluor. abs. 700 mm N2 fluor. NO (y system) FIG. 1 .-The effect of pressure broadening on light absorption and fluorescence. Nitric oxide pressure 2 mm.[To face page 302270 A 2370 8, blank before 3 psec 7 11 16.5 23-5 30 35 45 59 70 80 96 120 150 190 250 325 03 NO (1, 1) NO (0,O) NO ( 0 9 1 ) FIG. 2.-Formation and decay of vibrationally excited nitric oxide 2.2 mm NO with 430 mm N2, 1600 J flash energy.II II 2270 8, I II II 2370 A blank before 3 psec 7 psec 1 1 16.5 23.5 30 35 45 59 70 80 96 120 150 190 250 325 II II FIG. 3.-Fdrrnation and decay of vibrationally excited nitric oxide. 2.2 mrn NO with 430 rnm Kr, 1600 J flash energy.2270 A I 2370 A i before 3 dsec 7 11 16-5 23.5 30 35 45 59 70 80 96 120 150 190 250 cr, II II II II T n T NO ( 1 9 1 ) NO ( 0 , O ) NO (0, 1 ) FIG. 4.-Formation and decay of vibrationally excited nitric oxide. 2.25 mm NO with 540 mm He, 1600 J flash energy.1600 A + A I before 50 psec 07 1 070 1,o FIG. 7.-Formation of vibrationally excited CO. 5 nim NO, 100 mm CO with 650 mi N?.A. B. CALLEAR 31 and PI-0 for N2 = 4 x 10-7, the latter being accurate to about an order of magnitude. The value for the probability of self-relaxation by NO is in good argeement with that obtained by Bauer et aZ.2 by ultrasonic absorption and with the extrapolated shock- wave results of Robben. TABLE 1 mixtures (mm) 2 NO+ 600 N 2 2 NOf220 N2 5 NO+600 N 2 5 N0+560 N 2 1 N O + 6 W N 2 104 k (sec-1) 0.90 0.55 0.667 1 -72 1 -93 According to Robben,l the abnormally fast rate of relaxation by NO is due to sticky collisions and using an interaction potential of the N 2 0 2 molecule and the equations of Schwartz, Slawsky and Herzfeld,7 he obtained good agreement between theory and experiment.The ground state of nitric oxide is split into two spin orbit E- E 0.1 II a 0 x 2 0.05 2 f? W M n rcl 0 0 .- c) u 8 0 0 50 100 150 200 time (psec) FIG. 5.Variation of vibrationally excited nitric oxide molecules with time. 0 2 mm NO+6OO mm N2,16OO J 0 2 mm NO+220 lll~ N2,16OO J (> 1 ~III NOf600 mm N2, 1600 J 8 5 mm N0+600 N2,16OO J - - - 50 fll~ NO+457 mm N2, 900 J components, 2II3/2 and 2II112, the separation being 121 cm-1. Bauer et aZ.2 showed that spin orbit relaxation occurs about once in 16 collisions at 20°C and it seems likely that if the kinetic energy of the collision is high enough to cause vibration- translation relaxation, then the weak spin orbit interaction will be swamped and simultaneous spin orbit relaxation will occur.On this basis the most rapid process causing vibrational relaxation should be NO. X21T1,z(~ = 1)+NO . X2ITll2(u = 0)+2NO . X211312(~ = 0).32 VIBRATIONAL RELAXATION OF NITRIC OXIDE In this case the energy to be converted from vibration to translation is 1662 cm-1 instead of 1904 cm-1 for the fundamental frequency. This difference, however, is still not large enough to account for the fast relaxation observed using the theoretical equations.7 It would appear, therefore, that the sticky collision theory is the only possible interpretation of the abnormally fast relaxation, unless there are some special theoretical reasons why the occurrence of simultaneous spin orbit relaxation requires a modified treatment. QUANTUM YIELD FOR CONVERSION OF ELECTRONIC ENERGY TO VIBRATIONAL ENERGY Included on fig.5 is a curve showing the formation and decay of NO . X2rI(u = 1) with 50 mm of NO and 457 mm of N2. Under these conditions, the concentration of vibrationally excited molecules does not return to its initial very small value but remains at 0.025 mm for several msec and thereafter decays very slowly ; this is due to the increase in temperature caused by flashing. Under these conditions the fluor- escence is almost entirely quenched and the conversion of the electric energy to trans- lational energy heats the gas. Comparing with the absorption in pure NO at 20°C and applying the distribution law, the temperature is found to be 68°C; this is not precisely correct since, as previously mentioned, the conditions of pressure broadening are different in pure NO.The heat capacity of the gas is 0.02 cal/deg. (150 ml at 500mm pressure) and since one Einstein of radiation at 2270A is equivalent to 125 kcal, it follows that 7.6 x 10-6 Einstein have been absorbed which corresponds to the excitation of 0.95mm of NO. The maximum observed concentration of NO . X2II(u = 1) was 0.07 mm. From this the total number of vibrationally excited molecules can be estimated 3 by substituting an analytic function I = st exp (-p) for the variation of intensity of the flash with time and solving the linear differential equation, the first-order rate constant for the decay being determined with 50 mm of nitric oxide from the data of table 1. Thus it can be shown that under these condi- tions the total number of vibrationally excited molecules produced is about 0.7 mm.Thus, the quantum yield for the process NO. A2Z+(v = O)-++NO. X211(u = 1) is approximately unity. CHEMICAL REACTIONS I N FLASHED NITRIC OXIDE A very small fraction of the nitric oxide is photolyzed under the conditions de- scribed above, about 0.01 mm of NO being decomposed with a 1600 J flash. The decomposition can be almost entirely eliminated by use of an ammonia filter. It can be demonstrated experimentally by reflashing the same nitric oxide + inert-gas mixture several times that the decomposition has no detectable effect on the concentra- tion of NO. X2II(u = 1). Direct photolysis may occur due to absorption by the 6 bands or decomposition may arise by formation of NO.a4II and its reaction with other NO molecules. CATALYSIS OF VIBRATIONAL RELAXATION BY ADDED MATERIALS Addition of very small partial pressures of NH3 or H20 or a substantial pressure of CO, increased the rate of decay of vibrationally excited nitric oxide molecules. For H20 and CO, PI-0 was recorded as 7 x 10-3 and 2-5 x 10-5 respectively, by study- ing the decay after the fiash by plate photometry. The very fast rate of relaxation by water cannot be fully accounted for by means of a vibrational exchange mechanismA . B . CALLEAR 33 and is presumably due to a strong interaction between water and nitric oxide mole- cules. Fig. 6 shows the change in the rate of decay of NO. X2II(u = 1) in the presence of CO. Although the CO vibrational quantum is larger than that of NO, the prob- ability that the difference can be made up from kinetic energy is quite high even at 0 50 I00 150 time (psec) 200 FIG. 6.-Effect of carbon monoxide on the decay of vibrationally excited nitric oxide. (A) 2 mm NO with 600 mm N2.(B) 2 mm NO, 40 mm COY with 560 mm N2. Flash energy 1600 J. 20"C, so that the possibility of a vibrational exchange mechanism was investigated by carrying out some experiments in the vacuum ultra-violet region. Fig. 7 shows that on flashing a mixture of CO, N2 and NO, there was an increase in absorption of the CO (0,l) band ; the pressure of CO(u = 1) present initially was 8 x 10-4 mm and was increased by flashing to about 2-4 x 10-3 mni. Fig. 8 shows the variation of plate density with time of the CO (0,l) band. No change in absorption was observed in this wavelength region by flashing NO with N2, or CO with N2.In these vacuum ultra-violet experiments, the experimental arrangement was such that the temperature rise could not be measured satisfactorily because no " permanent " change in absorp- tion could be observed in the CO (0,l) band or any of the NO (u',l) bands due to a temperature rise. Consequently the quantum yield for formation of CO(u = 1) was not measured. The decay of the vibrationally excited CO is at least 50 times faster than it would be if the excitation was simply due to a temperature rise. VIBRATIONAL EXCHANGE PROCESSES The increased rate of decay of vibrationally excited nitric oxide molecules in the presence of CO is due to the exchange reaction. NO(U = l)+CQ(u = O)+NO(v = O)+CQ(v = 1).When vibrational equilibrium is reached in an equimolar mixture, there are 3.6 vibrationally excited NO molecules for each vibrationally excited CO molecule. Thus, if the CO is in large excess, the vibrational energy is drained away to the CO and at equilibrium, the concentration of NO . X2II(v = 1) is too small to be detected. The exchange reaction with N2 is very much slower because the energy discrepancy B34 VIBRATIONAL RELAXATION OF NITRIC OXIDE is larger. PI-0 = 0.4~ 10-6 recorded for relaxation of NO by N2 must be the probability of the exchange reaction, NO(v = 1)+N2(~ = O)+NO(V = O)+N,(u = l), since this value is very much larger than the theoretical probability of vibration- translation relaxation. It is difficult to obtain direct proof that this process occurs because absorption spectra of N2 can only be obtained at very short wavelengths.0.525 h + .* 0.55 5 0 0. c, e 0.575 0.5 1.0 1.5 2.0 2.5 time (msec) FIG. 8.-Photometer recording of the production and decay of vibrationally excited carbon monoxide. NO pressure = 5.0 mm CO pressure = 100 mm N2 pressure = 650 mm At vibrational equilibrium in an equimolar mixture of NO and N2, there are 9 vibrationally excited NO molecules for each N2 molecule. Since the maximum concentration of NO. X2II(v = 1) produced using either krypton or N2 as inert gases were about the same (fig. 2 and 3) and since the fluorescence is about the same intensity in either case, it follows that the NO. X2II(u = 1) is not in vibrational equilibrium with the N2 at short delay times, though equilibrium may be approached at long delay times. As pointed out above, the decay of NO.X2rl[(v = 1) in He, N2 and krypton is largely determined by self-relaxation by NO and in the presence of N2 the vibrational exchange process has only a minor effect. The reaction, CO(u = 1) + N2(u = O)+CO(U = 0) + N2(u = l), should be the fastest of the three exchange processes since it has the smallest energy discrepancy. It follows that the inclusion of CO in NO + N2 mixtures should cause vibrational equilibration of all three species. It can be shown 4 that the rate of decay of the vibrationally excited CO shown on fig. 8 is consistent with full vibrational equilibrium, the rate-determining step being Gaydon and Hurle 8 measured the vibrational relaxation times of N2 containing about 0.6 mm of CO, using chromium-atom reversal to measure the vibrational temperature of shock-heated gas.The relaxation times were shorter than in pure N2, which is consistent with the above discussion, since the translation-vibration relaxation time of CO is shorter than N2 and exchange of quanta occurs rapidly between CO and N2. NO(U = l)+NO(v = 0)+2NO(v = 0).A . B . CALLEAR 35 VARIATION OF THE RATE OF VIBRATIONAL EXCHANGE WITH ENERGY DISCREPANCY In order to determine the rate of vibrational exchange at exact resonance, Callear and Smith9 have studied the effect of nitrogen on the y fluorescence of NO. The fluorescence was excited with light from a xenon arc which produced molecules in the zeroth, first and second vibrational levels of the NO.A2Z+ state. The variation of the intensity in emission of the three progressions with inert gas pressure leads to values for the rates of vibrational relaxation in the electronically excited state. Thus, it can be shown that the number of collisions for vibrational exchange with N2 is approximately 500. It is seen from table 2 that the energy discrepancy is only 12 cm-1 in this case. TABLE 2 O'e (cm-1) 1904 2170 2359 2371 Fig. 9(a) shows the variation of the probability of vibrational exchange with the energy discrepancy. The number of collisions 2 refers to the exothermic exchange reaction in each case. A plot of log2 against energy discrepancy is linear, though 3 5 6 0 100 200 300 400 500 600 Av (cm-1) FIG. 9.Variation of the probability of vibrational exchange with energy discrepancy. this may be fortuitous since there may be a considerable experimental error in the point at 455cm-1. Curve (21) is obtained from the theory of Schwartz et aZ.7 using eqn. 65-9 and 65-14 given by Herzfeld and Litovitz.10 YO was taken as 3.6 A, 2" = 9 and the atomic weights of all the atoms were taken as 15. An ex- ponential function was fitted to the Lennard-Jones interaction potential at ro and at36 VIBRATIONAL RELAXATION OF NITRIC OXIDE the critical value of Y. The straight line c is an interpolation between curve (b) and the theoretical value at exact resonance. The agreement between experiment and theory is satisfactory. 1 Robben, J. Chem. Physics, 1959,31,420. 2 Bauer, Kneser and Sittig, J. Chem. Physics, 1959, 30, 11 19. 3 Basco, Callear and Norrish, Proc. Roy. Soc. A , 1961,260,459. 4 Basco, Callear and Norrish, Proc. Roy. SOC. A, 1962, 269, 180. 5 Callear and Norrish, Nature, 1960, 188, 5 3 . 6 Callear and Norrish, Proc. Roy. Soc. A, 1962, 266, 299. 7 Schwartz, Slawsky and Herzfeld, J. Chem. Physics, 1952, 20, 1591. 8 Gaydon and Hurle, Proc. Roy. Soc. A , 1961,262, 38. 9 Callear and Smith, to be published. 10 Herzfeld and Litovitz, Absorption and Dispersion of UZtrasonic Waves (Academic Press, New York and London, 1959).

ISSN:0366-9033    

DOI:10.1039/DF9623300028

出版商:RSC    

年代:1962

数据来源: RSC

摘要:

Some Aspects of the Theory of Vibrational Transition Probabilities in Molecular Collisions BY B. WIDOM Dept. of Chemistry, Cornell University, Ithaca, New York, U.S.A. Received 27th November, 1961 An exponential formula commonly used in the discussion of vibrational transition probabilities in molecular collisions is analyzed with the object of determining the extent to which it is dependent on specific (but questionable) assumptions about the form of the interaction energy. Using a principle of Landau, a very general expression is derived for the exponential appropriate to the one-dimensional case. It is shown that the leading term is very insensitive to the precise nature of the interaction, so that this term in its commonly used form has a great degree of generality. The correction terms in the exponential, on the other hand, depend sensitively on the assumed form of the interaction in the region of strong repulsion, so the corrections in current use must be considered very uncertain.1. INTRODUCTION It has in recent years become common practice to discuss vibrational transition probabilities in molecular collisions with the use of the formula, where P is the probability per unit time that the contemplated transition (a de-excita- tion) will occur ; p is the reduced mass of the collision partners ; A(> 0) is the mag- nitude of the change in the oscillator’s energy due to the transition ; B( >O) is the depth of the potential energy minimum in the interaction of the collision partners; II and kT have their usual meaning; and a is the range parameter which occurs when the interaction energy w as a function of distance x between centres of mass of the collision partners is represented by an exponential of the form w = A exp (- x/a) in the region of strong repulsion. The prz-exponential factor in P has been omitted from eqn.(1 . l ) because it is not involved in the present discussion. To obtain the form of the exponential in eqn. (1.1) appropriate to an excitation of the oscillator, rather than to a de-excitation, one has merely to replace A by - A . The first term in the exponential, as quoted in eqn. (l.l), is due to Landau and Teller,l while the remaining two terms, which are considerably smaller in magnitude than the Landau- Teller term, though still important, are due to Schwartz, Slawsky and Herzfeld,s 3 and will be referred to as the correction terms.Critical comparisons of eqn. (1.1) with experimental results have been made by Dickens and Ripamonti4 and by McCoubrey, Milward and Ubbelohde.5 The present discussion is concerned with the validity and degree of generality of eqn. (1.1). The Landau-Teller term and the correction term A/2kT were origin- ally derived 1 9 2 for one-dimensional systems with the explicit assumption of an exponential repulsion between the collision partners and, indeed, the former term still contains the range parameter characteristic of this potential. The correction term D/kT arose3 from a physical argument in no way dependent on an explicit choice of interaction potential, but the argument, though suggestive, is not obviously valid.37 P- exp [ - 3 ( 7 ~ ~ p A ~ a ~ / 2 h ~ k T ) ~ + A / 2 k T + DlkT], (1.1)38 VIBRATIONAL TRANSITION PROBABILITIES The analysis will be restricted to one-dimensional systems, but no special form for the interaction energy will be assumed until it is clear what the effects of such specialization are. The relative translational motion of the collision partners is essentially classical, so in the exponential expression for P the exponent, which arises from the translational motion, can be written as a sum of terms each classified according to its order of magnitude in A, and those terms which vanish as Pr-0 can be discarded. In this way there arise terms, such as the Landau-Teller term in eqn. (l.l), which arb large when ki is small, and also terms, such as A/2kT or D/kT, which are independent of A.By formally treating the energy A as an entity inde- pendent of fi, while letting h+O, the quantum nature of the vibrational motion and the essentially classical nature of the translational motion are both correctly taken into account. For a purely repulsive exponential interaction energy, the Landau-Teller term and the Schwartz-Slawsky-Herzfeld term A/2kT are exact, there being no other terms that survive when E+O. It will be seen that a slightly generalized form of the Landau-Teller term is still the correct leading term for a very general class of potentials. The correction terms, however, will be seen to be determined in general by certain detailed features of the interaction potential in the region of strong re- pulsion, and as these details are as yet unknown, both experimentally and theoret- ically, for real interactions, the correction terms must be considered to be in great doubt.The uncertainty is not over whether the Schwartz-Slawsky-Herzfeld terms A/2kT and D/kT should be included in eqn. (l.l), because it will be established that the former is surely, and the latter is probably, a correct %-independent term for general potentials ; but it will appear that these are by no means the only possible terms of this order of magnitude, and also that terms intermediate in magnitude between these and the leading (Landau-Teller) term may also be present. 2. TRANSITION PROBABILITY PER COLLISION Imagine an oscillator and an impinging particle in interaction in a one-dimensional system.Let be the displacement of the oscillator from its equilibrium position and let x be the distance between the centres of mass of the incident particle and of the oscillator (or of the molecule which contains the oscillator). The interaction energy W(x, c) between the two molecules, in the cases of interest here, may for all practical purposes be assumed linear in the oscillator co-ordinate.1 Thus, In any relevant initial or final state of an oscillator to which the theory might reason- ably be applied, the expectation value of is negligibly small, and it will be assumed here that it vanishes : The potential w(x) is then a typical intermolecular interaction energy, determining the relative translational motion of the two colliding particles both in the initial and final state of the oscillator.The forcef(x) perturbing the oscillator is not so familiar a quantity as is w(x). If W(x, r) were, say, a function of x-C alone (as is often as- sumed), then f ( x ) would be simply w'(x). More generally, however, W must be expected to be the sum of two functions, one of x and one of x-5 (imagine the oscillator to be contained within a large molecule), so that in reality w(x) and&) are quite independent. Alternatively, one can describe the situation by saying that the force which determines the relative translational motion, - w'(x), does not consist only of the reaction, -f(x), of the force on the oscillator, but also of an <5> = 0. (2.2)B. WIDOM 39 additional force which would act between the two colliding particles even if no force acted on the oscillator itself.However, while not much is known about f ( x ) , its effects appear only in the pre-exponential factor in eqn. (l.l), and since the present discussion is restricted to the exponential factor alone, the nature of f(x) is irrelevant. That the magnitude of the transition probability per unit time depends almost entirely on the potential which determines the relative translational motion, and hardly at all on the perturbing force, is the most interesting point in the theory. The probability p that the oscillator will undergo the transition i-+j due to collision with the incident particle may be calculated reliably by the method of distorted waves.6 Alternatively, it may be calculated by the method of perturbed stationary states,6 with the additional assumption that the perturbation of the oscillator energy levels and wave functions by -f(x)c is adequately given by first-order perturbation theory. In either case, it follows from eqn.(2.1) and (2.2) that the probability p is given by the following formula, essentially due to Zener : 7 8pt2. - rJ f (x)G(E, x)G(E +A, x)dxI2, = h2 JE(E +A) - co where ( i j is the matrix element of 5 taken between the unperturbed initial and final states of the oscillator, where E is the smaller of the initial and final relative trans- lational energies in the collision (so that E+ A is the larger), and where for any positive energy E the wave function G(E, x) is the well-behaved solution of r2E+E-w(x) 2p ax2 1 G(E, x) = 0, which is normalized so that as ~ 4 ~ x 3 , where w(x) vanishes, G(E, X) - cos (J2pExlft + 6) (2.9 with 6 a real number which depends on E, and on the form of w(x), but which is independent of x.Either of two forms of the intermolecular potential w(x) might reasonably be envisioned for this one-dimensional system. (i) The interaction becomes infinitely repulsive at a finite value of x (which, without loss of generality, can be taken to be x = 0), as would be the case if w(x) were proportional to a negative power of x. (ii) The interaction is finite for all finite x, but becomes infinitely re- pulsive as x-+ - co, as would be the case if w(x) were of the form A exp (- xla). In case (i), the wave functions G in the integral of eqn. (2.3) are to be taken identically 0 for x<O (so that the integration can be restricted to the interval 0, co) and the well-behaved solution of eqn.(2.4) for x>O is that which vanishes at x = 0. In case (ii) the well-behaved solution of eqn. (2.4) is that which vanishes as x-+ -a. The crucial step in the discussion is now the recognition of the important general principle of Landau,s according to which, if the translational motion is essentially classical, then the integral in eqn. (2.3) is f(x)G(E, x)G(E + A, x)dx - exp JW -(E +A)x’(w)dw- where x(w) is the function inverse to w(x), and x’(w) is its derivative. The pre- exponential factor, containing A only multiplicatively, is not relevant here and has40 VIBRATIONAL TRANSITION PROBABILITIES been omitted. Eqn. (2.6) represents the asymptotic evaluation of the integral on the left-hand side, as A+O. It is here that one sees that the exponential depends only on the potential w(x) and not at all on the forcef(x).The power and generality of Landau’s principle, in the form of eqn. (2.6), can be tested by applying it to three special cases for which the integral on the left-hand- side is known exactly : (9 w(x) = A exp (- x/a), f(x) = B exp ( - x/b). This is the exponential interaction; in the special case b = a the exact result is due to Jackson and Mott 9 and has been the basis of most previous discussions of vibrational transition probabilities. There is no difficulty in extending the exact result to general b, and as R+O the integral on the left-hand-side of eqn. (2.6) becomes where the pre-exponential factor contains k only multiplicatively and has been omitted. This is precisely the exponential given by the right-hand-side of eqn.(2.6). That the exponential depends on a but not on b is due to the fact that it depends on w but not onf. (ii) w(x) = D[exp (-x/a)-2 exp (-x/2a)], f(x) = Aw‘(x) ’-“his is the Morse potential, for which the exact evaluation of the left-hand-side of eqn. (2.6) is due to Devonshire.10 Letting 5-0 and omitting the pre-exponential factor, Devonshire’s result reduces precisely to that calculated from Landau’s principle, viz. , where for any positive E, +(E) = JETD arc tan J T D - 3 In (1 + E/D). (iii) W ( X ) = A/X2, f(x) = B/x2. This is the inverse square potential; the exact value of the integral 11 yields for its exponential factor, in the limit A+O, (1 + A/E)- dPA/2E2 again in agreement with Landau’s principle.Thus, whether w(x) becomes infinitely repulsive at x = --GO or at a finite value of x, and whether it is purely repulsive or also partly attractive, eqn. (2.6) gives the correct result. The transition probability per collision, which is essentially the square of the quantity in eqn. (2.6), may, therefore, be considered known explicitly for any intermolecular potential w(x). If in eqn. (2.6) one makes the assumption that A<E, and so discards all but the leading term in A, the result is identical to what would be obtained from the so-called semi-classical theory 12 if in the latter theory the collision integral were evaluated in the asymptotic limit li+O. This was clearly recognized by Landau and Teller 1 and, indeed, was the basis of their analysis.Thus, the semi-classical theory contains nothing which is not already contained in eqn. (2.6). Furthermore, it is capable of yielding only the leading term, but not the correction terms, in expressions such as that of eqn. (1.1).B. WIDOM 41 3. TRANSITION PROBABILITY PER UNIT TIME The de-excitation probability per unit time P is obtained from the transition probability per collision p(E) by * P = J p ( ~ ) z ( ~ ) d ~ , (3.1) 0 where z(E)dE is the number of collisions per unit time suffered by each oscillator, in which the initial relative translational energy is between E and E+dE. The integral is then to be evaluated in the asymptotic limit fi+O. The exponential part of p(E), as given by eqn. (2.6), and the factor exp (- E/kT) in z(E), are the only factors in the integrand which contribute to the exponential part of P, and the result, from eqn. (2,3), (2.6) and (3.1), is that where E* is the solution of co (n !)- 'A"F(")(E*) = B/,fGkT, n = 1 and where F(n)(E) is the nth derivative of the function F(E) defined by F(E)=J&- * X'(W) - E (3.3) (3.4) To evaluate these quantities one must know the detailed behaviour of x'(w) for large w, that is, the detailed behaviour of the interaction potential w(x) in the FIG.1.-The function x(w) is the inverse of the intermolecular potential w(x), and its derivative x'(w) is shown as a function of w. Curve a represents the case of a purely repulsive interaction while curve b, with two branches, represents the case in which the interaction energy has an attractive region of depth D.region of strong repulsion. Such knowledge would lead to asymptotic expressions for F(E) and its derivatives at high energies, and this is what is required since E* is large. Two possible forms of the function x'(w) are shown qualitatively in fig. 1, curve a representing the function when the interaction is purely repulsive, and curve b (which consists of two branches) representing the function when w(x) has an attractive region of depth D.42 VIBRATIONAL TRANSITION PROBABILITIES All that one can say with reasonable assurance about the behaviour of x’(w) for large w is that it is of the form x’(w) = -~aw-~[I+g(w)] (a>0, N>1) (3.5) where For example, the exponential potential w = A exp ( - x / a ) is the special case N = 1, g(w) = 0, while the inverse power potential w = Ax-m is the special case a = m-lAlim, N = 1 + l/m, g(w) E 0.Note that as far as Nis concerned, the exponential potential is an inverse power potential in which the power is infinite. So long as the power is large, whether it be finite or infinite, N in eqn. (3.5) will be close to, and not less than, unity. This observation is important in establishing the generality of the leading term in the exponential in P. It follows from eqn. (3.2) (3.5) that, except for terms in the exponential which vanish as B-+O, g ( 0 ) = 0. P - exp [ - E*/kT + (,/aE)AF(E*) + A/2kT] with the first two terms themselves given by I‘(N++) J2TaAkT 1/(NS3) h 1 - E*JkT + (J&i/ft)AF(E*) = - + correction terms (3.7) The leading term in the exponent is that given explicitly on the right-hand-side of eqn. (3.7).It is the generalization of the Landau-Teller term to arbitrary potentials, and reduces to it for the exponential potential. Since, as noted above, N is close to 1 for all strong repulsions, whether exponential or not, the leading term in eqn. (1.1) remains nearly correct for all realistic cases. The situation as regards the correction terms is quite different. The Schwartz- Siawsky-Herzfeld term A/2kT appears explicitly in eqn. (3.6) as a correct A-inde- pendent term for the most general potential, but additional terms of the same (and, perhaps, even greater) magnitude are contained among the correction terms on the right-hand-side of eqn.(3.7), and these terms require for their evaluation a detailed knowledge of the function g(w) in eqn. (3.9, that is, a detailed knowledge of the inter- action potential at high energies. Foi example, if g(w)-b/w for large w, then the complete exponent in P consists of the leading term plus the correction terms A/2kT-b/NkT, so that in this case there is an %independent correction in addition to A/2kT. There is, of course, no Teason whatever to assume that g(w) behaves like l/w for large w ; for a Morse potential, or for a Lennard-Jones 212-n potential, g(w) behaves like l/w* for large w, and then correction terms appear which, as far as their order of magnitude in li is concerned, and probably their numerical mag- nitude as well, are intermediate between the leading term and the additional fi- independent corrections. The correction term D/kT in eqn.(1.1) might appear in the following way. Suppose the lower branch of curve b in fig. 1 is represented by xf(w) = -a(w+ D>-N, which is qualitatively correct. Then g(w)- -ND/w for large w, s0 from the dis- cussion of the preceding paragraph it follows that the correction terms are precisely A/2kT+D/kT, and no others, just as in eqn. (1.1). This argument makes it highly likely that D/kT is also a correct h-independent term in the general case, but all the ditficulties concerning the probable presence of other terms remain.B . WIDOM 43 The conclusion one must draw from the foregoing analysis is that the leading term in the exponent of P is known in great generality, but that the corrections are still very uncertain. This research was supported by the U.S. Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command. 1 Landau and Teller, Physik. 2. Sowjet., 1936, 10, 34. 2 Schwartz, Slawsky and Herzfeld, J. Chem. Physics, 1952, 20, 1591. 3 Schwartz and Herzfeld, J. Chem. Physics, 1954, 22, 767. 4 Dickens and Ripamonti, Trans. Faraday Soc., 1961, 57, 735. 5 McCoubrey, Milward and Ubbelohde, Trans. Faruday SOC., 1961, 57, 1472. 6 Mott and Massey, The Theory of Atomic Collisions (Oxford University Press, Oxford, 1949), 7 Zener, Physic. Rev., 1931, 37, 556. 8 Landau and Lifshitz, Quantum Mechanics (Pergamon Press, London, 1958), pp. 178-183. 9 Jackson and Mott, Proc. Roy. SOC. A, 1932,137,703. 10 Devonshire, Proc. Roy. SOC. A, 1937, 158,269. 11 Widom and Bauer, J. Chem. Physics, 1953, 21, 1670. 12 Zener, Physic. Rev., 1931, 38, 277. 2nd ed., chap. 8.

ISSN:0366-9033    

DOI:10.1039/DF9623300037

出版商:RSC    

年代:1962

数据来源: RSC

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Interpretation of Rate Experiments With Resolved Quantum Levels BY TUCKER CARRINGTON National Bureau of Standards, Washington 25, D.C. Received 24th January, 1962 It is now possible to study the steady-state fluorescence and chemiluminescence of small molec- ules in resolved quantum levels, or to observe the time history of the vibrational or rotational relaxation of molecules suddenly produced in a non-equilibrium distribution. These experiments require interpretation in terms of rates or probabilities of vibrational or rotational energy exchange involving specified quantum states. If it is assumed that the state of a molecule after a collision depends only on its state immediately before the collision, and not on its previous history, or on the time at which the collision occurs, the sequence of collisions suffered by one molecule becomes a Markov chain, and the appropriate mathematical formalism may be directly applied. This estab- lishes a relationship between the observable transition rates and the more abstract concept of probability of transition per collision.The mathematical results are illustrated in discussing certain experiments . The rise of non-equilibrium molecular and chemical kinetics in theoretical and more recently experimental studies has given insight into the processes of energy transfer between molecules in resolved quantum levels. Where we once measured rate constants as a function of temperature, thus assuming equilibrium, and later measured relaxation times, often in complex systems where their physical sig- nificance is small, we are now entering a period when several experimental techniques make possible the observation and eventually the calculation from experimental data of transition rates between resolved rotational and vibrational quantum levels.This focuses attention on the statistical concepts which relate the rates or proba- bilities of transitions involving specific quantum states to the observed distribution of molecules among these states. N 0 N- EQUILI B R I U M EXPERIMENT S Before discussing statistical models and current experiments, we may review briefly earlier work on non-equilibrium molecular and chemical kinetics. The study of unimolecular reactions was perhaps the first clear entrance of non-equilib- rium effects, particularly in the fall-off of the rate at low pressures, and in the negative activation energy for the reverse reaction at these pressures.Another important area of study has been the so-called relaxation times for rotational or vibrational energy transfer, measured by the techniques of sound dispersion and absorption, shock waves, and impact tube. In a system of many available energy levels, a single relaxation time has significance only when the system is close to equilibrium, and in general bears no simple relationship to transition rates linking particula levels.1 In a series of experiments, McKinley, Garvin and Boudart,2 Garvin,3.4 and Cashion and Polanyi 5 3 6 have studied spectroscopically infra-red chemiluminescence from reactions of the type A+ BC-+AB + C in which AB may have vibrational energy amounting almost to the entire exothermicity of the reaction.In these experiments one resolves vibrational and rotational structure, and so one can study the population of these levels as a function of pressure and diluents. From these data, crude 44T. CARRINGTON 45 calculations of “ initial distributions ” or rates of population of the various levels by the reaction itself have been made, thus “ subtracting ” from the observed distribu- tions the effects of collisions following the initial reaction. The experiments have been designed primarily to get information about this initial distribution, character- istic of the chemical reaction itself, rather than about the rates of dissipative processes tending to establish equilibrium distributions of rotational and vibrational energy in the molecules after they are formed.In most spectroscopic studies of non- equilibrium systems, the separation of these two processes is extremely difficult. Experiments carried out by Norrish’s group have dealt with the same problem, and have recently been reviewed.7 Flash-lamp techniques are used, and the time resolution thus obtained eliminates some of the averaging over the distribution of free times between collisions which is involved in steady-state experiments. The use of optical absorption spectroscopy allows the determination of population in the ground state as well as in excited states of the molecules observed. In most respects, however, the interpretation of the steady-state experiments and flash-lamp experiments is very similar.8 MOLECULAR COLLISIONS AS A MARKOV CHAIN We now consider the statistical treatment of molecules which can exchange rotational and vibrational energy with a heat bath.In particular, we consider a system of identical, non-interacting molecules, capable of existing in a large number of accessible energy levels. These are dilutely dispersed in an inert gas, a heat bath of fixed temperature, with which they can exchange energy. A given molecule, in the course of its collisions with inert molecules, undergoes random transitions among these energy levels, a sort of random walk, which, however, has the property that transitions toward the levels most populated at equilibrium are somewhat more frequent than those in the opposite direction. This random sequence of transitions can be described as a Markov process because the state of a molecule after its next collision depends only on its state just before that collision, and not on its past history.DISCRETE AND CONTINUOUS TIME DEPENDENCE In connection with observation of transitions from level j to i in a given experi- (a) the rate of transitions, wij, transitions/sec per molecule ; (b) an arbitrarily assumed collision rate, z, collisions/sec per molecule, corres- ponding to an arbitrarily assumed collision cross-section ; (c) the probability of transition per collision, pij ; (d) the probability of transition in time interval 0 to t, Xij(t). mental system, four concepts describing the transition are of interest : We consider first only two levels, and then extend these ideas to the matrices in- volved when many levels are treated.If we consider only one kind of transition, we measure a mean rate w in the ex- perimental system. It is generally desirable to express this rate in a dinensionless form, by comparing it with another rate. For this purpose we assume an arbitrary collision cross-section and from this and the density of heat bath molecules we calculate the mean collision rate z. The quantity w/z has some meaning as a probability of transition per collision when it is less than unity, but it must be kept in mind that some collisions in which the distance of closest approach is equal to, or less than, that corresponding to the arbitrarily chosen cross-section will not lead46 RATE EXPERIMENTS WITH RESOLVED QUANTUM LEVELS to transition, and some more distant collisions will induce transitions.The probability of transition in a given collision is never precisely unity. If we ask for the probability of a transition in a given time interval t, the obvious answer is X(t) = pzt, the probability of transition per collision times the number of collisions in the time interval t. This will have meaning as a probability only when pzt< 1. The simplest form for X(t) for arbitrary t is obtained by assuming an ex- ponential distribution of free times between effective collisions. The treatment then resembles that of an optical absorption coefficient. The fraction of molecules which undergo a transition (are absorbed) in time t is X(t) = 1 -exp (- wt). For small t this approaches X(t) = wt = pzt, as we should expect.The simple ideas just discussed are easily extended to the general case of many available energy levels. We define the matrix A having as elements the rates wij, and express A as zQ, where z is a constant scale factor representing a maximum collision rate, and the off-diagonal elements of Q are qij = W i j / z . Since transitions j to j are unobservable in the present model, the diagonal elements of Q are deter- mined by conservation of number of molecules. This gives q j j = -2qij so that Q has column sums zero. The problem now is to establish relationships between the matrices A (hence Q), P and X(t). We again set pij = WQ-/Z, but now z must be chosen large enough so that &j< 1 for all j. The diagonal elements are then defined by pij = 1 - 2 pij = 1 +ajj/z.This definition makes qgj = pij for i#j, and in fact Q = P-1. We now consider the relation between the matrices X(t) and P. The elements Xij give the fraction of molecules in state i at time t, given that all molecules were in state j at time zero. It represents the observed time behaviour of the system, and can also be considered as the probability of finding a molecule in state i, given that it was initially in state j . This matrix X(t) has been frequently discussed,*. 9 and is given by X(t) = exp (At). Substituting P for A and expanding in the limit t+O gives X(t)-,l+z(P-1)t. If this limiting expression may be used for t as large as t = l/z, we have X(l/z)wP. This says that the probability of any transition during a period equal to the mean free time is approximately equal to the probability of that transition per collision.This approximation gets better and better as we increase z, since the Wij are fixed and the qij thus gets smaller. Increasing z, however, means including relatively distant encounters as collisions, so that the concept of collision covers more different types of encounters and is less well defined. Consider again a number of molecules all of which are in the same quantum level at time zero. They undergo collisions and collision induced transitions randomly in time. We may investigate the applicability of a model in which each molecule makes collisions separated by the time interval l/z, i.e., at a regular rate. Knowing the matrix P, we can calculate the distribution of molecules among the available levels after any number of collisions.If x(n) is a vector in which each element xi(n) gives the fraction of molecules in level i after n collisions, where x(0) is the initial distribution. It is of interest to compare this result with the correct treatment based on collisions occurring at random intervals. In this case The difference between the two models lies in the fact that all the molecules will i i # j i#5 x(n) = P"x(O), (1) x(t) = exp (At)x(O), (2)T. CARRINGTON 47 not make the same number of collisions in any given time interval. We may co,m- pare (1) and (2) by setting t = n/z and expanding : (3) x(n) = P”x(0) = (l+Ajz)”x(O) = (l+nA/z+n(n-1)A2/2z2+ . . .)x(O), x(t = nlz) = exp (An/.) = (1 + nAlz + n 2A2/2z2 + .. .)x(O). (4) The two expressions then tend to agree in the limit t = 0, particularly when z is large and the elements of A are small. For the real physical systems we are dis- cussing, the matrix P has the property lim Pn = POo, so that the distribution ap- proaches a limiting equilibrium value as time, or number of collisions, increases. The two forms (3) and (4) therefore agree also in the limit t+m. The assumption that each molecule makes exactly the average numberlof collisions in any period of time may lead to significant errors in some cases, particularly in steady-state experiments. 10 A detailed comparison of random and regular collision models treating fluorescence in comets has been made elsewhere.11 n--tc;o FORMAL DESCRIPTION OF EXPERIMENTS INVOLVING RESOLVED QUANTUM LEVELS The concepts just presented can be used to give a concise formal description of models which are idealizations of experiments of the types described in the pre- ceding references.2-7 Discussing again a heat bath in which molecules of interest are dilutely dispersed, suppose that we have in some volume under observation an initial distribution xo, a vector in which each element xp gives the fraction of molec- ules initially in level i. Suppose that these molecules are removed by processes unspecified, except that the rate of removal of molecules in level i is zi per second per molecule.We provide further that molecules are constantly introduced into the system at a total rate c per unit volume, with a fraction xi(0) going into level i.This influx of molecules begins suddenly at time zero and is steady thereafter. We wish to investigate the time dependence and steady state of this system. Denoting the total number of molecules in unit volume by n, we have dx . CXi(0) 1 = CA..x.-z.x.+ -, n U J 1 1 dt dxldt = (A - Z)x + (c/n)x(O), or, in matrix notation, in which 2 is a diagonal matrix having elements zi.12 In the steady state we must have (c/n)zxi(0) = cln = ‘&xi. i i (7) Putting this value of c into (5) gives an equation for calculating the steady-state distribution, in which the superscript s labels the steady-state distribution and the matrix z is given by Both A and z have the property that all column sums are zero, hence the determinant of (A+z) is zero and a solution to (8) exists, determined by the requirement ZXP = 1.(A + ;)x(’) = 0 zi j = z jxi(o) - Zi6i j . (8) (9) * 148 RATE EXPERIMENTS WITH RESOLVED QUANTUM LEVELS If all the zi are equal, eqn. (8) reduces to where z is l/zi. If in eqn. (8) one assumes that every collisional transition probab- ility is proportional to the corresponding radiative transition probability, with always the same proportionality constant k, each term is multiplied by a factor (1 + kn), where n is the density of molecules in the heat bath. This cancels everywhere, and there is no effect of inert gas pressure.5 If x(~) is known from observation and a transition probability model is assumed, eqn. (8) can be written in the form (1 - zA)dS) = x(O), (10) ~ ( 0 ) = (Z - A)x'"/? (8') convenient for calculating x(0).Here Z is the average, Z = Zi&). i In terms of the steady-state solution, the differential eqn. (6) can be written, dx/dt = (A - Z)(X - x(')), (x - x@)) = exp (A - z)t(x" - x(~)), and its solution is as is readily verified by differentiation. The eigenvalues of A are not greater than zero 9 and those of Z are just the zt, so that the matrix exponential 13 goes to zero as t becomes large, and of course approaches unity as t goes to zero. This ensures that the solution (12) has the necessary properties in these limits. If there is no steady input, and no removal of molecules from the system, we have the simple relaxation equation The form of (1 2) indicates that the " quenching " processes zi which remove molecules from the system will speed up the attainment of the steady state.When Z is large, there will be little time for the relaxation processes described by A so, for example, there may be relatively little rotational or vibrational energy transfer during the observed lifetime of the molecules of interest. The model just discussed is applicable to the experiments of Polanyi596 and Garvin 3 , 4 if the energy levels involved are excited vibrational levels only. The " quenching " processes zi are then all those which produce transitions to the ground vibrational level. Since these molecules are unobservable and we ignore their col- lisional re-excitation to higher levels, they are essentially removed from the system. The model also describes in an idealized way, the flash-lamp experiments of Norrish's group.7 Here in the simplest case, the steady input is zero, 2 = 0, and all levels are included in the distributions of interest. The initial distribution xo relaxes to thermal equilibrium according to eqn.(1 3). The matrix equations just presented are, of course, entirely equivalent to the cor- responding systems of scalar equations describing the same model, and the matrix formulation introduces no new physical content. It is, however, useful in several ways. It divides the complexity of the problem into two parts, each of which may be more easily understandable and tractable than the sea of algebraic equations with which one is otherwise flooded. The first part is the formulation of the problem in matrix notation. In many cases it is possible to do this directly, without resort to the much more cumbersome scalar notation.The second part is the solution of the equations by the techniques of matrix algebra. Here the " book- keeping " required to keep track of the many energy levels is done automatically, and the many mathematical results concerning the properties of matrices can be applied. The matrix formulation is also convenient for use with high-speed electronic x = (exp At)x". (13)T. CARRINGTON 49 computers, for example, in predicting experimental results which would follow from some assumed set of transition probabilities. With the experimental pre- cision now available, it is in general not possible to calculate transition rates from experimental data, especially when transitions are not restricted to nearest neighbour levels? In fact, the observed distributions are quite insensitive to changes in the transition rate values.This is because there are in general a great many ways of going from statej to some other state k in, say, 5 transitions. Many of these possible sequences of transitions will not involve some particular transition pq, even when p or q may lie between j and k. Any particular transition probability ppq will not play a strong role in determining the results. What is observed is an average of arbitrarily many Markov chains of variable length, but usually short, since the ob- served distributions are generally far from equilibrium. TREATMENT OF EXPERIMENTAL RESULTS The treatment just outlined can be applied to the interpretation of several experi- ments involving observation of collision effects on molecules in resolved quantum levels.We now discuss two such examples. VIBRATIONAL DISTRIBUTION I N CHEMILUMINESCENCE In the steady-state experiments of Garvin and of Polanyi, the main interest is in deriving the relative rates of population of the several vibrational and rotational levels produced by the reaction under study. To do this, one has to allow for, or in some sense subtract, the effects of subsequent collisions tending to bring the ex- cited molecules into equilibrium with the heat bath. To do this, one has to assume a set of rate coefficients or transition probabilities for the transfer of vibrational or rotational energy among all the levels involved. The treatment developed in the preceding section can be used conveniently to compare results obtained with several different transition-probability models.As an illustration of this, we consider the work of Garvin on the reaction H + 0 3 = 0 2 + OH, wher6 the piime indicates vibiational excitation. We consider three sets of transition probabilities. In model a, we use radiative transition prob- abilities, as calculated by Garvin, and assume that the collisional transition prob- abilities are proportional to these. This model involves some relaxation of the harmonic oscillator selection rule Av = 1, but nearest neighbour transitions are still greatly predominant. In model b, nearest neighbour transitions are still favoured, but not so much as in a. The absolute values are chosen as to give, for each upper level, the same total probability of leaving that level in b as in a.In model c, transi- tions from a given upper level to all lower levels are equally probable, and the total transition probability from each upper level is normalized as before. Applying these three models to Garvin’s observed vibrational distribution 4 gives, using eqn. (S‘), three “ estimates ” of the input distribution, x(O), shown in fig. 1. The very drastic change in the transition-probability model from a to c has only the effect of smoothing out the x(0) distribution somewhat. This is strong indication that if accurate transition probabilities were known, the input distribution derived from them would not be very different from those shown in the figure. It should be emphasized that the model used here ignores collisions between two vibrationally excited OH molecules, such collisions being rendered negligible by the heat bath.Broida 14 has proposed that collisions of this type, OH’ + OH’-+OH* + OH, pro- ducing an electronically excited OH molecule, are important in these experimental systems.50 RATE EXPERIMENTS WITH RESOLVED QUANTUM LEVELS I I I MONOCHROMATICALLY EXCITED FLUORESCENCE We mention now a group of experiments in which the excitation mechanism is clearly known and fixed, and interest centres on collision processes by which rotational and vibrational energy is exchanged with a heat bath. These are steady-state fluor- escence experiments in which a single quantum level of rotation and vibration is excited by illumination of a gas with a well-chosen sharp emission line from an arc or discharge tube, which overlaps an absorption line of the gas.One then observes fluorescence from this initial level and from neighbouring levels populated from it by collisions. Many experiments of this type have been done,ls-ls but they have not allowed detailed interpretation. An experiment of this type with OH has recently been reported,lo and one is now in progress with NO. I 0 1 2 3 4 5 6 7 8 9 vibrational quantum number FIG. 1.-Relative rates of production of vibrationally excited OH molecules in the reaction HfO3 -+ OH’+02, calculated from experimental results of Garvin et al., ref. (4). Three different vibrational transition probability models are used. a consists of the radiative transition probabilities, and strongly favours nearest neighbour transitions.In c jumps of any Av are equally likely, and b is intermediate. The curve with no maximum is the observed distribution, ref. (4). The monochromatically excited fluorescence of NO was studied by Kleinberg and Terenin,lg using the cadmium line at 2144A. They obtained electronic and vibrational energy transfer collision efficiencies, but did not resolve the rotational structure. Experiments of this type are now being repeated with adequate rota- tional resolution.20 In the upper electroiiic state 2C the level v’ = 1, K’ = 13 is excited, with about two-thirds of the molecules having electron spin in the opposite direction to that of the rotational angular momentum. For NO+NO collisions, rough preliminary results give collision efficiencies of 1.3 for electronic quenching, 0-5 for vibrational energy transfer u’ = 1 to v’ = 0, and 1.5 for rotational energy transfer out of K’ = 13.The two spin states in rotational levels other than 13 seem to be roughly equally populated. One also observes fluorescence from the levelT . CARRINGTON 51 v’ = 0, populated by vibrational energy transfer collisions. A study of the rotational structure of this level should give information about the simultaneous changes in rotational and vibrational quantum number. Our observed efficiency of quenching from the first vibrational level is about 3 times of that found by Basco, Callear and Norrish.21 Addition of argon markedly increases the amount of rotational energy transfer in the v’ = 1 level.This is presumably because argon is much less effective as a collision partner than NO in removing electronic and vibrational energy, so molecules live longer in v’ = 1 and undergo more rotational transfer collisions. Addition of chemically inert gases thus allows one to change the effective time-scale for rotational relaxation. Concerning the rotational-energy distributions, one can test any transition- probability model by using the model and the known initial level to compute the distribution to be expected. The inverse problem, computing a transition-probability model from the experimental results, is in general much more difficult.8 It is a pleasure to acknowledge helpful conversations with David Garvin. The work was supported by the Aeronautical Research Laboratory, Air Force Research Division. 1 Shuler, Physics Fluids, 1959, 2, 442. 2 McKinley, Garvin and Boudart, J. Chem. Physics, 1955, 23, 784. 3 Garvin, J. Amer. Chenz. Soc., 1959, 81, 3173. 4 Garvin, Broida and Kostkowski, J. Chem. Physics, 1960, 32, 880. 5 Cashion and Polanyi, Proc. Roy. SOC. A, 1960, 258, 529, 570. 6 Cashion and Polanyi, J. Chem. Physics, 1961, 35, 600. 7 Basco and Norrish, Can. J. Chem., 1960, 38, 1769. 8 Carrington, J. Chem. Physics, 1961, 35, 807. 9 Montroll and Shuler, Adu. Chem. Physics, 1958, 1, 361. 10 Carrington, 8th Symp. Combustion (1960), p. 257. 11 Carrington, Astrophys. J., May, 1962. 12 A similar formulation has been discussed by Wilson, J. Chem. Physics, 1962, 36, 1293. 13 Bellman, Introduction to Matrix Analysis (McGraw Hill Book Co., New York, 1960). 14 Broida, J. Chenz. Physics, 1962, 36, 444. 15 Wood, Phil. Mag., 1918, 35, 236. 16 Polanyi, Can. J. Chem., 1958, 36, 121. 17 k n o t and McDowell, Can. J.Chem., 1958, 36, 114, 1322. 18 Durand, J. Chenz. Physics, 1940, 8, 46. 19 Kleinberg and Terenin, Doklady Akad. Nauk S.S.S.R., 1955, 101, 445, 1031. 20 unpublished work with H. P. Broida. 21 Basco, Callear and Norrish, Proc. Roy. Suc. A , 1961, 260,459.

ISSN:0366-9033    

DOI:10.1039/DF9623300044

出版商:RSC    

年代:1962

数据来源: RSC

摘要:

Collisional Energy Transfer between Electronic and Vibrational Degrees of Freedom BY P. G. DICKENS, J. W. LINNETT AND 0. SOVERS Inorganic Chemistry Laboratory, Oxford Received 18th January, 1962 A formal theory €or the collisional transfer of energy between electronic and vibrational degrees of freedom is outlined. Calculations are made with an ultra-simplified form of the interaction potential for the systems : (a) Wls) + H(1 s) -932s) + H(I $1, (b) Hg(3P1)+ BC +Htd3po) + BCt, (c) Na(2P) + BC -+Na(2S) + BCt. It is demonstrated that on the basis of the simplified interaction potential assumed, the exchange transfer between electronic and vibrational degrees of freedom is generally a very inefficient process, except for a near coincidence of vibrational and electronic energy levels.Conditions under which such resonance effects can be important are discussed. The object of this paper is to investigate the part played by the uptake of vibra- tional energy in the quecching of electronic excitation energy and in related processes. In particular, it is proposed to examine the conditions under which resonance effects such as those reported by Zemansky 1 might be expected. The quenching reactions and Hg(3P1) + BC+Hg(3Po) + BCt Na(2P) + BC-+Na(2S) + BCt where t represents a vibrationally excited molecule, are taken as examples. 1. GENERAL FORMULATION The co-ordinates employed in the description of a collision of an atom A with a diatomic molecule BC are as follows. r is the distance between the centres of mass of the colliding systems, r1 repre- sents an electronic co-ordinate of the atom, q is the normal co-ordinate for the vibra- tion of BC, do is the equilibrium distance between B and the centre of mass of BC.The axis of BC is assumed to lie along r with B nearest to A. For simplicity, the electronic states of the atom are assumed to be spherically symmetric and are described by the wave functions &(rl). The wave equation describing the collision process is where 4(rl) and ~ ( q ) are the corresponding wave functions for the internal motions concerned, Eo is the total internal energy of the isolated systems initially and H,, and H, are the corresponding Hamiltonians. Hereafter, $(rl,q) is written for 52P. G. DICKENS, J . W. LINNETT A N D 0. SOVERS 53 $(rl)~(q). p is the reduced mass of the colliding systems = mAmBC/(mA+mBC), and uo is the relative velocity of approach. V(r,rl,q) is the total interaction potential for the two systems averaged over the electron co-ordinates of BC.Rotation of BC is ignored. For a change in internal states of 0-n, a solution of eqn. (1) is sought of the form such that Ro+ exp (ikor cos O)+fo(6,$) exp(ikor)/r, r+ca and (4) Rn+f n(e9 4) ~ X P ( iknr)ir9 r+m in order that Ro should represent an incoming plane and outgoing elastically scattered spherical wave, and R, should represent an inelastically scattered spherical wave.2 ki is the wave number 2npvi/h, where vo and un are the relative velocities of approach and departure respectively (subsequently k is written for k, to simplify notation). The differential inelastic cross-section for the transition O-+n, on (6,00), is given by an(07u0)do = (k/ko) Ifn(e,6> l2 do, ( 5 ) where dw is an element of solid angle.The total inelastic cross-section, a,(u,), is given by integration of (5) over the surface of the unit sphere. Substitution of (3) into (1) leads to the set of equations where n Vin = V(r,rl,q)$i$nd?, J (7) and J. . . dz’ represents integration over the internal co-ordinates rl and q. Two common approximate methods available for the solution of the eqn. (6) are (a) the Born approximation, and (b) the method of distorted waves? (a) is suitable for very high energies of approach and (b) is valid when the off-diagonal elements Vff are < the diagonal elements Vti. A great simplification is found if V(rl,q,r) can be written in the form V(r)V(rl)V(q), in which case Yon can be written as uonV(r), where U0n is JV(r1)V(q)$o$,dz’. For actual molecules, the true V(rl,q,r) is an impossibly complicated function to work with, save in the very simplest cases,3 and progress can only be made by finding a simple but fairly accurate approximate form for V(rl,q,r). 2.A SIMPLE APPROXIMATE FORM FOR THE INTERACTION POTENTIAL For close distances of approach Voo(r) may be expected to have the form Vo exp (- ar),4 where a is a constant. It is proposed to use for an approximate interaction potential (between atom A and atom B) a function of the form V = Vi (1 + prl)exp( - ad), (8) where d is the (instantaneous) distance between atoms A and B and p is a parameter with dimensions of reciprocal length and magnitude of ~ 1 0 8 cm-1.The physical model underlying this assumption is that around atom A is pictured a spherical shell of electron density, of radius r1, the position of which produces a modulation of the interaction potential for a fixed value of d. d can be written in the form,54 COLLISIONAL ENERGY TRANSFER d = r - d o - x ~ , where XB is the vibrational amplitude of B and is equal to Aq where A is constant and q is the normal co-ordinate. Hence, eqn. (8) can be rewritten V = Vo (1 + prl)exp( - ar) exp(aAiq) = V(r)V(q) V(rl). (9) Voo x Kn w V, exp( - ar), and Vonm V, uonexp( - ar). (10) As a consequence of the form of (9) it follows that Hence in this model there can be no crossing of potential-energy curves and thus it should only be applicable to systems in which the two electronic levels concerned are well separated in energy from one another and also from any other neighbouring states.The formal analogy with vibration-vibration energy transfer is immediate and the relevant formula of Witteman 5 and Herzfeld and Litovitz 6 can be applied. To test the validity of the form of the interaction potential (8), the collisional excitation of the 2s level of a hydrogen atom by another ground-state hydrogen atom was considered. Bates and Griffing3 have applied the Born approximation using the exact interaction potential, in the calculation of the total inelastic cross-section a,(vo) for the process exchange effects being neglected. Use of the Born approximation with the inter- action potential (8) leads to an expression for the differential inelastic cross-section of H( 1 s) + H( 1s) + H(2s) + H( 1 s), an(8,vo) = (k/ko)(2p/h2)2u&, where K = I K I, and K = k-ko is the vector difference between the final and initial wave vectors k and ko.That is, K2 = ki+k2-2kko cos 8. One finds 5 for the total cross-section Conservation of energy requires that where AE = En- Eo, and En and EO are the total internal energies of the n and 0 states respectively. To compare the two solutions, the parameters VO, a, and uon must be fixed, where kg - k2 = 2pAE/ii2 Bates and Griffing's calculated results are plotted in fig. 1, together with those derived from function (12). The value of a was chosen so that the maxima of the two curves coincide (the position of the maximum of (12) depends only on a).The product ( Vouon)2 was chosen to give the same values of a, at the maximum. The correspond- ing values were a = 3 x 108 cm-1, (VO~O,)~ = 1.22 x 10-18 erg2. If p-1 be taken as 1 A and 40 and $, are assumed to be hydrogen 1s and 2s functions respectively, then u&wO.O9 ; hence for internal consistency V; must be taken as 1.4 x 10-17 erg2. The absolute values of a and VO so chosen appear reasonable in comparison with other intermolecular force constant data.4 Of greater importance, however, is that the form of the interaction potential (9), which allows a separation of variables, leads to a reproduction of the salient features of electronic excitation in atomic col- lisions, namely, the sharp initial rise in the inelastic cross-section with increasingP .G. DICKENS, J . W. LINNETT AND 0. SOVERS 55 energy of approach at low incident energies followed by a much slower fall-off in cross-section at high energies.2 In the remaining sections we shall not be concerned with the absolute value of the electronic matrix element in U0n and this term can be looked upon as a scaling factor when considering the relative changes in cross-section for a given electronic energy change. loglo [incident energy (keV)] FIG. 1 .-Plot of cross-section against incident energy for collisional excitation of the H(2s) level : (I) present work, (11) Bates and Griffing (ref. (3)) 3. APPLICATION TO THE QUENCHING OF MERCURY RADIATION BY DIATOMIC MOLECULES For the low energies of approach encountered under gas kinetic conditions the Born approximation is not applicable.The method of distorted waves is more suitable since Von = Vo u,,exp(-ar)< Voo,T/nn. Making use of the analysis given by Witteman,s an expression for the total inelastic cross-section is found to be m o,(v,) = ( 4 n / k k ~ ) ( 2 p / ~ 2 ) 2 ~ & C (21 + 1)A&, l = O where 00 A,, = J v(r)FlkoFZkdr* Flko and Flk are auxiliary functions which satisfy the equation with the associated boundary conditions, [d2/dr2 + k2 - 1(1+ l ) / r 2 - (2p/fi2)Vnn(r)]Flk = 0, Flk--+ sin (kr-&En+tjlk) and Fzk = 0 at r = 0. r-+ 03 I is the usual angular momentum quantum number of the central field problem, and V(r) = Gn(r) = Vo exp( - ar).56 COLLISIONAL ENERGY TRANSFER An exact analytical expression can be found for the first term in the summation in (13).A convenient approximate expression for the complete sum as given by Witteman is where O0 = 27rko/a, 8 = 2nk/a and u$, and u$k are the matrix elements of Y(r1) and V(q) between the initial and final electronic and vibrational states respectively.* Conservation of energy requires ii2(k; - k2)/2p = AEvib-AEel = AE, where A&ib is the vibrational quantum gained or lost by the diatomic molecule and AEel is the electronic quantum lost or gained by the mercury atom. AE is the net energy transferred to translation. Negative values of AE correspond to an increase in relative translational energy. The total inelastic cross-section appropriate to a particular temperature is ob - tained by averaging a,(uo) over a Maxwellian distribution of velocities of approach, that is, for net deactivation Q* = +(p/kT)2 Jmcn(vo)v: exp ( - pvg/2kT)dvo0 (17) 0 The related cross-section for activation NUMERICAL CALCULATIONS Expression (17) was used in the calculation of the inelastic cross-section for the process, where * refers to electronic excitation and t to vibrational excitation ; AE is the net energy transferred to translation. Qd was calculated for a hypothetical series of diatomic molecules BC of the same mass but with different vibrational quanta.The electronic quantum removed from the excited mercury atom was taken to be 0.219 eV (23P1+23Po), and the vibrational quantum taken up by BC was allowed to vary in the range 0.219fO-2 eV. There appeared to be no sound a priori method for the estimation of the range of the interaction potential, a-1, and calculations were therefore made for several values of a employing the sort of magnitude used in the calculation of vibrational relaxation times in gases.6 A reduced mass of 25 a.m.u.was used since this value approximates those appropriate to the Hg + CO, N2, NO systems. Q d was calculated for 300°K and the integral in eqn. (17) was evaluated numerically. Hg** +BC*Hg* +BCt +AE, (18) Fig. 2 shows a plot of loglo Qi against AE, where and AE is the resonance defect previously defined. Qi is measured in units of 10-16 cm2. The quantity (u$,)2 will remain effectively constant for a series of molecules with the same mass but different vibrational frequencies ; (u$:)2 will * Current work on the numerical solution of eqn.(15) suggests that the approximation (16) considerably underestimates the value of a,(v,).P. G. DICKENS, J . W. LINNETT AND 0. SOVERS 57 vary with vibrational frequency - l / h which, at least in the range near AE = 0, is a much slower vaiiation than that of Q&. Hence, the general features of fig. 2 may be used in a discussion of the variation of Q d with the resonance defect for a one-quantum vibrational change in BC. The form of the curve of log Q d against AE is typical of a resonance effect, a consequence of which is that electronic-vibration exchange, involving the excitation of a single vibrational quantum, is an extremely improbable process except for those cases in which the vibrational quantum involved lies very close in energy to the AE (eV) FIG.2.-Plot of relative cross-section Qi against resonance defect (AE) for Hg+BC. electronic quantum removed (AE,l/hc = 1770 cm-1). Those exchanges are most probable for which a minimum of energy is transferred to translation. This con- clusion is in accord with experimental experience.8 The order of efficiencies of the diatomic molecules, N2, CO and NO found in the quenching of 3P1-+3Po mercury radiation is NO> CO> N2.8 This is understandable since this is the order of in- creasing resonance defect (?NO = 1904 cm-1, 9 ~ 0 = 2170 cm-1, C N ~ = 2360 cm-1). The relative efficiencies read off from fig. 2 (for a = 5 x 108 cm-1) are QNO = 26Qco = 250Q~,, those found experimentally are QNO = 6Qco = 129 QN,. Values for other a’s are shown in table 1. The calculated result is very sensitive to the value of a chosen.Quenching of the same radiation by inert gas atoms would be predicted by this calculation to be quite negligible since such atoms have no energy levels close in value to the electronic quantum removed and hence all the excitation energy must be transferred to translation (AE = -0.219 eV). This conclusion is again in agreement with experiment; inert gas atoms are invariably found to be inefficient quenchers .4 To illustrate the relative efficiencies of multi-quantum transfers of vibrational energy, calculations were made for a series of hypothetical diatomic molecules with fundamental vibration frequencies of 500, 1000, and 1500 cm-1 respectively. Relat- ive total inelastic cross-sections for reaction (1 8) were computed corresponding to the58 COLLISIONAL ENERGY TRANSFER uptake of 1, 2, 3, etc., quanta of vibrational energy.S.H.O. matrix elements (u&~)Z were calculated. The results are shown in table 2. It is clear that there are two opposing effects : (a) a multiple quantum jump, which leads to a small AE, provides a favourable translational factor Qi, but (b) a multiple quantum jump is associated with a much smaller vibrational matrix element than is a single quantum process. In general, factor (a) appears to be the more important; the maximum cross-section TABLE 1-RELATIVE CROSS-SECTIONS FOR THE PROCESS Hg(3P1)+ BC+Hg(3Po)+ BC? molecule loglo(u~b)z loglo Q~(AZ) loglo ~ i ( U ; ; i ; ) ) ' ( ~ z ) rel. cross-section 3 N2 - 2-64 - 3.50 -6.14 1 co - 2.60 - 1.85 - 4.45 49 NO - 2.55 + 1.20 - 1.35 6 . 2 ~ 104 5 N2 - 2.20 - 1.50 - 3.70 1 co - 2.16 - 0.55 - 2.71 9.8 NO - 2.10 + 0-80 - 1-30 250 7 N2 - 1 *90 - 0.70 - 2.60 1 co - 1.87 -0.15 - 2-02 3.8 NO - 1.81 + 0.60 - 1.21 25 TABLE 2.-RELATIVE CROSS-SECTIONS, Qi, FOR TRANSFER OF A H@P1 + 3 P ~ ) ELECTRONJC QUANTUM TO THE TZTH VIBRATIONAL LEVEL OF THREE HYPOTHETICAL MOLECULES AE, eV vibrational frequency of molecule, cm-1 500 0 -0.219 1 -0.157 2 - 0.095 3 - 0.033 4 + 0.029 5 + 0.09 1 lo00 0 - 0.21 9 1 - 0.095 2 + 0.029 3 +0.153 1500 0 -0.219 1 - 0.033 2 +0*153 p = 25 a.m.u., loglo Qi(A2) - 9-50 - 6-90 - 3.90 - 0.35 + 0.35 -2.15 - 9.50 - 3.90 -415 - 9.50 - 0.35 -4.15 + 0.35 a = 5 A-1, 0 - 9.50 - 1.47 - 8.37 - 3.24 - 7.14 - 5.19 - 5.54 - 7.26 - 6.91 - 9.43 - 11.58 0 - 9-50 - 1.77 - 5.67 - 3.84 - 3.49 - 6.09 - 10.24 0 - 9.50 - 1.95 - 2.30 - 4.20 - 8.35 T= 300°K.will derive from a process in which AE is minimized. However, for the three vibrators considered, the smallest AE in the particular case results from the transfer of respectively 3, 2, and I vibrational quanta. The relative cross-sections for these most favourable transfers fall in the ratios 1 : 1.1 x 102 : 1.7 x 103. That is a single quantum jump which can also make AE very small will produce the largest cross- section. Anharmonicity will tend to make the differences between the vibrational matrix elements for single and multi-quantum jumps smaller (see 5 4) and thus cause the requirement of near resonance to become more rather than less important.P . G . DICKENS, J .W. LINNETT AND 0. SOVERS 59 4. QUENCHING OF SODIUM RADIATION BY DIATOMIC MOLECULES The quenching reaction Na(2P) + BC+Na(2S) + BC', (19) differs from (18) principally in that a much larger electronic quantum (2 eV) is involved. From the discussion in 5 3, it is clear that use of the simple interaction potential (8) will lead to a very small cross-section for this process. This is because any vibrational transition which makes AE sufficiently small, and hence Q; large, will be offset by a very small vibrational matrix element for a multi-quantum jump (a 0-7 vibrational transition would be needed with N2 to bring AE-0). The results of a calculation displayed in table 3 demonstrate this. The calculation was made TABLE 3 RELATIVE CROSS-SECTIONS, FOR TRANSFER OF A Na(2P-tzS) ELECTRONIC QUANTUM TO THE IZTH VIBRATIONAL LEVEL OF A N2 MOLECULE n loglo Q;(u ::)' (A') Morse S.H.O.AE, eV log10 ~J(8.z) S.H.O. Morse 0 1 2 3 4 5 6 7 8 9 - 2.095 - 1.804 - 1.517 - 1.234 - 0.954 - 0.677 - 0.404 -0.135 +0*131 + 0.393 - 18.10 - 15.95 - 13.65 - 11.40 - 8.85 - 6-25 - 3.25 + 0-20 + 0.40 - 2.30 0 - 204 - 4.38 - 6-90 - 9.54 - 12.28 - 15.10 - 17.99 - 20.93 - 23.93 0 - 2.02 - 3.79 - 5.40 - 6.88 - 8.26 - 9.57 - 10.81 - 11.98 - 13.11 - 18.10 - 17.99 - 18.03 - 18.30 - 18.39 - 18.53 - 18.35 - 17.79 - 20.53 - 26.23 - 18.10 - 17.97 - 17.44 - 16.80 - 15.73 - 14.51 - 12.82 - 10.61 - 11.58 - 15-41 for a Na+N2 collision at 2500°K with a taken as 6 x 108 cm-1. Vibrational matrix elements were calculated both for a S.H.O. and for a Morse oscillator. The effect of anharmonicity is to increase uZb for the 0-7 transition by a factor of about 104, but even so the resulting cross-section would be extremely small ( m 10-10 A2).This result is not in agreement with experiment since N2 is found to be a fairly efficient quencher of Na(2P-G) radiation.ss9 A reason for this discrepancy is not hard to find. The form assumed for the interaction potential in (8) requires V&m Vnn. Hence for all values of r, where En and EO are the total internal energies in the final and initial states respec- tively. Hence, a large cross-section can only be found within the framework of this assumption for cases in which the vibrational energy taken up, preferably in 0-1 quantum change, is rn Suppose on the other hand VOO and Vnn can differ considerablyover some range of Y, and suppose further that En - EO in the absence of any vibrational change is large, that is AEel>hvvib.Then for some region of Y, (and hence for which En-EowO). may become very small, equal to 6, say. If 6=hvvib, an efficient transfer involving the uptake of a single vibrational quantum would then be possible, although En - Eo> hvvib. Such a situation is equivalent to the near crossing of the potential energy60 COLLISIONAL ENERGY TRANSFER curves corresponding to the initial and final electronic states. That is for cases in which VOO and Vnn are very different, no correlation is to be expected between the efficiency of electronic-vibration energy transfer and the near coincidence of a vibra- tional energy level with the energy difference between initial and final electronic states.This correlation might be expected only where VOO- Vnn for all r and hence where there is little tendency towards a crossing of the potential energy curves. Such a situation seems to arise in the quenching of H~(~PI-'~Po) radiation but not for the quenching of Na(ZP4S) radiation. A comprehensive theoretical treat - ment must be based, therefore, on a detailed knowledge of the relevant potential energy surfaces for each collision pair. These data are rarely available. The simple interaction potential used here, though incomplete, does enable one to relate the particular case of electronic-vibrational energy transfer to general scattering theory and to look at the factors affecting efficiency of transfer in these terms. We wish to thank the Royal Society and the Imperial Chemical Industries for providing calculating machines. P.G.D. and O.S. thank the Pressed Steel Company and the National Science Foundation (U.S.A.) respectively for the award of Post- Doctoral Fellowships. 1 Zemansky, Physic. Rev., 1930, 36, 919. 2 Mott and Massey, The Theory of Atomic Collisions (O.U.P., 1949). 3 Bates and Griffing, Proc. Physic. SOC. A , 1953, 66, 961. 4 Hirschfelder, Curtiss and Bird, The Molecular Theory of Gases and Liquids (Wiley, New York, 5 Witteman, J. Chem. Physics, 1961, 35, 1. 6 Herzfeld and Litovitz, The Absorption and Dispersion of Ultrasonic Waves (Academic Press, 7 Jackson and Mott, Proc. Roy. SOC. A, 1932, 137,703. 8 Laidler, The Chemical Kinetics of Excited States (O.U.P., 1955). 9 Clouston, Gaydon and Hurle, Proc. Roy. SOC. A, 1959, 252, 143. 1954). New York, 1959).

ISSN:0366-9033    

DOI:10.1039/DF9623300052

出版商:RSC    

年代:1962

数据来源: RSC

摘要:

Intermolecular Transfer of Vibrational Energy BY J. D. LAMBERT, A. J. EDWARDS, D. PEMBERTON AND J. L. STRETTON Physical Chemistry Laboratory, University of Oxford Received 15th January, 1962 Measurements of ultrasonic dispersion have been made in mixtures of pairs of polyatomic gases, which have near-resonant vibrational energy levels. Single dispersion was observed for mixtures of CH30CH3 with CC12F2, CH3Cl and SFs. Double dispersion was observed for mixtures of C2F4 with CHF3, CF4 and SF6 and for a mixture of C2I& with SF6. All mixtures showed a linear dependence of reciprocal relaxation time on molar composition, extending over the whole com- position range. The results are interpreted on the hypothesis that rapid transfer of vibrational energy from molecule A to molecule B in a “ complex collision ” is followed by deactivation of B in col- lisions involving the normal transfer between vibrational and translational energy.There is little experimental evidence about the direct transfer of vibrational energy between molecules in collision in the gas phase. Measurements of vibra- tional relaxation times in pure gases by ultrasonic and other techniques have been mainly concerned with vibrational-translational energy transfer. On theoretical grounds it would appear unlikely that there is any appreciable hindrance to a reson- ance transfer of a quantum of vibrational energy from one molecule to another in a homo-molecular collision, and chemical spectroscopic evidence 1 confirms that equilibrium distribution of vibrational energy between molecules of a single species is very rapidly attained.The role of the “ complex collision ”, in which a vibrational quantum is transferred froin molecule A, plus or minus the necessary increment of translational energy to excite a different vibrational quantum of molecule B, has been investigated by ultrasonic techniques only for rare cases, where energy transfer between the different vibrational modes of a single species of polyatomic molecule involves a measurable relaxation time (CH2C12,2 and SO2 3). The complex collision can also play an important role in the overall vibrational- tianslational energy transfer process in heteromolecular collisions. The possible energy transfer processes are here : A* + A-+A+A (vib-+trans) (1) A*+B-+A+B (vib-trans) (2) A* +B-+A+B* (vibjvib) (3) B*+B-+B+B (vibjtrans) (4) B* +A-+B+A (vib-trans).(5) If A is a relaxing molecule with a low transfer probability, and B a non-relaxing molecule or a molecule with a much higher transfer probability, a rapid process (3) may be followed by a rapid process (4) or (5), giving rise to a greatly enhanced vibration-translation transfer probability for molecule A. Such a mechanism may be one of the factors responsible for the powerful “catalytic” action of certain foreign molecules in promoting energy transfer.4 Rapid vibration-vibration energy transfer would be expected to occur between molecules with near-resonant vibra- tional frequencies. This paper describes ultrasonic dispersion measurements on a number of mixtures of polyatomic gases, which fulfil this criterion to varying degrees.6162 VIBRATIONAL ENERGY TRANSFER The usual expression for the composite relaxation time 18 of a mixture of a re- laxing gas A with another gas B, where only processes (1) and (2) operate, is : where x is the mole fraction of component B, and BAA and PAB are the relaxation times corresponding to the transfer probabilities in processes (1) and (2) respectively. When complex collisions occur, the validity of this expression will depend on the relative rates of the three processes, (3). (4) and (S), iiou7 involved in the vibrational deactivation of A. If (3) or (5) is the late-controlling step, eqn. (6) will still be valid, giving a linear dependence of 1/P on x. If (4) is the rate-controlling step, this will no longer be so, since this rate is proportional to x2.Such non-linear dependence has been demonstrated for the variation of the relaxation time of 0 2 (v = 1580 cm-1) with small additions of H20 (v2 = 1595 cm-I), which was in- vestigated by a shock-tube method.5 (Linear dependence was found for mixtures of 0 2 with D20 (v2 = 1178 cm-I), where no vibration-vibration transfer would be expected, so that only processes (1) and (2) operate.) Since the detailed dependence of relaxation time on composition is an important factor in elucidating the mechanism of energy transfer, all the systems described below have been investigated over as wide a range of composition as possible. For mixtures where both components show relaxation which is experimentally measurable, a double relaxation phenomenon would be expected 4 with two distinct relaxation times, PA and PB, corresponding to the vibrational deactivation of A and B, respectively, such that 1 1-x x + -Y P A BAA PAB _ - -- and 1 x 1-x -- --+----, PB PBB PBA (7) where PBB and PBA are relaxation times corresponding to processes (4) and (5), respectively.It is difficult to obtain experimental resolution of relaxation times differing by a small factor, or in systems where the specific heat contribution of one component heavily outweighs that of the other, and previous attempts to demonstrate this phenomenon have not been conclusive.6?7 Double relaxation is established for four of the systems described below. EXPERIMENTAL Measurements were made in two acoustic interferometers, which have already been described in essentials.8 The lower frequency interferometer was used with a quartz oscillator of frequency 200kc/sec and was operated at room temperature.The higher frequency interferometer was used with a 4 Mc/sec quartz crystal, with nodal mounting of the type described by Arnold, McCoubrey and Ubbelohde,g driven by a Clapp oscillator, and was operated at 25°C. Measurements were made at varying pressures between 0.1 and 1 atm. Gas mixtures were made by techniques described previously,lO and were intro- duced to the evacuated interferometer by a Topler pump. The compositions of mixtures were checked from time to time by mass spectrometer analysis or by a Janak gas chrom- atography apparatus. Gas-imperfection corrections for pure gases were made as described previously, using data listed for each gas below.The second virial coefficient of a mixture was calculated from the expression, B = (1 - x ) ~ B A + 2 ~ ( 1 -x)BAB + x 2 B B .J . D. LAMBERT, A . J . EDWARDS, D. PEMBERTON AND J . L. STRETTON 63 BA and BB are the second virial coefficients of the pure components. BAB, representing the heteromolecular interaction, was calculated by the methods of Hirschfelder, Curtis and Bird 11 where appropriate data were available, or by the methods of Guggenheh and McGlashan.12 Relaxation times were calculated by fitting theoretical dispersion curves to the experimental points, which were usually at least 15 to 20 in a set. The necessary specific heats were calculated from the spectroscopic data listed below, except for CH30CH3, where the vibrational assignment is uncertain, and calorimetric values 13~14 were used.The static specific heat of a mixture was taken as and the relaxing contributions calculated from the appropriate vibrational frequencies. (1 - x>c; + xc; , MATERIALS C2H4 : two different specially pure cylinder samples were used, which showed relaxa- tion times in excellent agreement. Gas imperfection, Berthelot equation. Spectroscopic data.15 SF6 was obtained from a cylinder. Mass-spectrometer analysis showed no detectable impurity. Gas imperfection.16.17 Spectroscopic data.18 C2F4 was a sample supplied by I.C.I. Mass spectrometer analysis showed no de- tectable impurity. Gas imperfection.24 Spectroscopic data.25 CHF3 was a sample used previously.19 Gas imperfection.19~ 20 Spectroscopic data.21 CF4 was a sample used previously.19 Gas imperfection.22 Spectroscopic data.23 CH30CH3 was obtained from a cylinder.Mass spectrometer analysis showed no CH3Cl was obtained from a cylinder specified to contain less than 200 p.p.m. by weight detectable impurity. Gas imperfection, Berthelot equation. Spectroscopic data.26 dimethyl ether as impurity. Gas imperfection.27~ 28 Spectroscopic data.23 CC12F2 impurity. M 4 0 - --- 8 - & 0 L? was obtained from a cylinder specified to contain less than 70 p.p.m. by volume Gas imperfection.29 Spectroscopic data.30 I I I I L 6 . 0 6.5 7.0 7.5 loglo c f p ) (c sec-1 atm-1) FIG. 1.-Dispersion curve for mixture of CF4 with 60.2 mole % c2F4. 0, experimental values ; -- , theoretical curve for double relaxation process ; P(CF4) = 5.2 x 10-8 ; P(C2F4) = 1.42 x 10-8 sec ; - - -, theoretical curve for single relaxation process, involving total vibrational energy of both molecules.RESULTS The reciprocal of the measured relaxation time is shown plotted against molar For the mixtures involving composition for the various mixtures in fig. 2-8.64 VIBRATIONAL ENERGY TRANSFER CH30CH3, whose relaxation time is so short as to lie outside the range of experi- mental measurement, only single relaxation was observed, corresponding to the VibrationaI relaxation of the second component. For all the other mixtures, double relaxation was observed : a specimen dispersion curve for CF4+ C2F4 is shown in - 800 Y ! 0.5 1.0 mole fraction SF6 c2H4 SF6 FIG.2.-Reciprocal relaxation times for CzHd+ SFs mixtures. mole fraction C2F4 CHF3 C2F4 FIG. 3.-Reciprocal relaxation times for CHF3 + C2F4 mixtures. fig. 1. Separate relaxation times relating to each of the components were calculated 2,s described previously.3 All the mixtures show a linear plot of l/p against x, in accordance with eqn. (7) and (8), enabling values of PA*, PAB, ,!?BB and PBA to be calculated. The relaxation times for the pure gases are in general agreement with previous measurements in this and other laboratories. The value for CH3C1 differsJ . D. LAMBERT, A. J . EDWARDS, D. PEMBERTON, AND J . L STRETTON 65 from that of Fogg,lg who used a sample known to contain 2 % of CH30CH3; it is in good agreement with measurenients of Edmonds and kamb.31 Pure CH30CH3 appears to be showing incipient dispersion at the highest value of frequencylpressure attained (107.65 c sec-1 atm-1) and a minimum mole fraction C2F4 cF4 c2F4 FIG.4.-Reciprocal relaxation times for CF4+C2F4 mixtures. mole fraction C2F4 value of the 800 ,600 7 i3 U > ,400 200 FIG. 5.-Reciprocal relaxation times for SF6+ c2F4 mixtures. possible relaxation time is estimated as 2.5 x 10-9 sec. Fig. 2-5 are accompanied by energy level diagrams to show the possible vibration-vibration transfers. The upper harmonics of energy levels are included in the diagrams where appropriate as dotted lines. C66 VIBRATIONAL ENERGY TRANSFER DISCUSSION It is convenient to discuss first the mixture, C2H4 + SF6, where complex collisions are unlikely to play an important role in facilitating energy transfer.The relaxation times for pure C2H4 and SF6 do not differ greatly, and it will be seen from fig. 2 that '4 I J 0.5 I. mole fraction CH30CH3 = -t.- FIG. 6.-Reciprocal relaxation times for CClzF*+ CH30CH3 mixtures. mole fraction CH30CH3 n 400 ----- =%=Ox - 1 CH3Cl CH30CH3 FIG. 7.-Reciprocal relaxation times for CHjCl+ CH30CH3 mixtures. C2H4 + SF6 collisions are of roughly the same efficiency as C2H4 + C2H4 collisions in deactivating C2H4, while substantially more effective than SFs + SF6 collisions in deactivating SF6. The energy level diagram shows that there is an easy transferJ . D. LAMBERT, A. J . EDWARDS, D. PEMBERTON AND J . L. STRETTON 67 between the 810 cm-1 mode of C2H4 and the 775 cm-1 mode of SF6, but this could not be responsible for increasing the efficiency of deactivation of SF6 in heteromole- cular collisions above that of the self-deactivation of C2H4.In terms of the reaction scheme given in eqn. (1) to (5), however rapid process (3) may be, the overall relaxa- tion rate cannot be faster than either process (4) or process (5). The explanation of the enhanced heteromolecular collision efficiency for SF6 would seem to lie in the normal vibration-translation transfer by process (2) being facilitated by the rela- tively low mass and steep intermolecular potential of the C2H4 molecule. The observed increase in collision efficiency by a factor of 5-9 is possible on these grounds. 0- 5 mole fraction CH3OCH3 FIG. 8.-Reciprocal relaxation times for SF6+ CH30CH3 mixtures.Similar theoretical considerations predict that the efficiency of deactivation of C2H4 in collisions with SF6 would be lower than in self-collision. The fact that the experimentally measured efficiency shows little change may be interpreted in terms of a complex collision involving very rapid energy transfer from the 810 cm-1 mode of C2H4 to the 775 cm-1 mode of SF6, followed by deactivation of the SF6 molecule by a combination of processes (4) and (5). This might be expected to lead to a non- linear plot in fig. 2, as the more efficient heteromolecular collisions (process (5)) would play a larger role in the mixtures richer in C2H4. Unfortunately, the close- ness of the two relaxation times makes it impossible to resolve them for the mixtures richer in SF6 and to decide whether the observed line is, in fact, curved.The little change observed in the heteromolecular deactivation efficiency of C2H4 is, however, in accord with this closeness. All the remaining mixtures show a much larger difference between the relaxation times of the two pure components. If A is the gas with the longer relaxation time (lower efficiency of deactivation), the values of PAB are consistently smaller than those of PAA to an extent which is unlikely to be accounted for by difference in molecular mass or repulsion potential. Indeed, for two mixtures, CHF3+C2F4 and CF4+ C2F4, the effect of mass would operate in the reverse direction. The energy level diagrams show that, in all cases, easy vibration-vibration transfers are available, so that a mechanism involving complex collisions seems likely to be responsible for the enhanced efficiency of heteromolecular collisions in deactivating A.For the three68 VIBRATIONAL ENERGY TRANSFER mixtures where the relaxation time of the other component B can be observed, the values of ~ B A fall a little more than those of PBB. (The appearance of fig. 3, 4 and 5 is misleading in this respect, as the ratio of the two l/p intercepts, which is the relevant factor here, is nut proportional to the slope of the line, and the variation in relaxation time of component B appears exaggerated in comparison with that of A.) This means that heteromolecular collisions are slightly less efficient than self- collisions in deactivating B ; the effect is of a size which can probably be accounted for by differences in intermolecular repulsion potential and/or mass.There is no case for invoking complex collisions to account for a decrease in efficiency of deactivation, and a simple vibration-translation transfer (process (5)) is an ap- propriate mechanism for the deactivation of B. The values of PYA given in table 1 are calculated on the assumption that the whole of the vibrational energy of B relaxes through the lowest mode by this mechanism : p7A = (C7/Cr)pBA, see below. It now remains to discuss the details of the complex collision mechanism re- sponsible for the heteromolecular deactivation of A. The three mixtures involving C 2 F 4 as the B component (fig. 3, 4 and 5) are the easier to interpret, and may be TABLE 1 RELAXATION DATA - 35 CHF3 C2F4 0.519 0.164 0.119 2.49x 10-7 1.61 X 10-9 2.30X 10-9 1.84X 10-9 1.4 1-25 0 CF4 C2F4 0.584 0.164 0.104 5.14X 10-7 1.61 X 10-9 2.46X 10-9 2.97X 10-9 1-5 0.53 - 5 SF,j CzF4 0.310 0.164 0.070 1.96x 10-7 1 6 1 X 10-9 3.20X 10-9 10.3X 10-9 2.0 0.31 +17 - 1.79X 10-8 - c2H4 SF6 0.277 0.310 - 6.37X 10-8 1.96X 10-7 - CCIzF2 CH30CH3 0.189 0.249 0.1 14 1.69 X 10-8 < 6.3 X 10-10 4.26X 10-10 - >0*7 - - 1 1 CH3CI CH30CH3 0438 0.249 0.204 9.41 x 10-8 < 6-3 x 10-10 1-57X 10-9 - >2*5 - + 18 SF6 CH3OCH3 0.310 0.249 0.086 1.96x 10-7 < 63X 10-10 1.13X 10-9 - >1.8 - - 35 considered together.For the C H F 3 + C 2 F 4 mixture, there is a resonance vibration- vibration transfer between two 507 cm-1 modes, while for the S F 6 + C 2 F 4 and C F 4 f C 2 F 4 mixtures the values of Av for vibration-vibration transfer are only 17 and 5 cm-1 respectively.Process (3) would therefore be expected to be extremely rapid in all three cases, so that the rate-controlling step would be the faster of (4) or (5). Predominance of (4) might be expected to give rise to a non-linear plot of 1/p against x, but the measured rates of (4) and (5) differ so little that the observed linear dependence seems justified if both processes are operating. (This is in sharp contrast to the 0 2 + H 2 0 mixture mentioned above,s where the efficiency of vibra- tional deactivation of H 2 0 in self-collisions is some 103 times that in H 2 0 + 0 2 collisions.) If the mechanism is as suggested, the measured relaxation time PAB corresponds to relaxation of the whole of the vibrational specific heat contribution of both molecules (C$+ Cs), via the lowest mode of molecule B, whose contribution is C?.It has been shown that for self-collision of polyatomic molecules, where the whole of the vibrational energy of the molecule relaxes through the lowest mode, the effective relaxation time of this mode is given by provided it is assumed that energy transfer between modes is very rapid.32 ByJ . D. LAMBERT, A. J. EDWARDS, D. PEMBERTON AND J. L. STRETTON 69 analogy the effective relaxation time for energy transfer via the lowest mode of B in the heteromolecular process should be given by and the value of pfB would then be expected to lie between byB and pBA, since processes (4) and (5) control the rate. Values of fltB calculated by eqn.(9) are shown in table 1, together with values of p7B and p:A and the ratios of fltB to these. It will be seen that for the CF4fC2F4 and SF6fC2F4 mixtures the value of fl+B lies in between PyB and fl?* in accordance with expectation. For the CHF3+C*F4 mixture, fl?B lies slightly above both flyB and flyA ; in view of the complexity of the mechanism and the many assumptions involved, this is not unsatisfactory agreement. Interpretation of the remaining three mixtures (fig. 6, 7 and 8) is less satisfactory, as the full dispersion zone of the B component CH30CH3 is not accessible to the present apparatus, and only the relaxation of the A (less efficient) component can be observed in the mixtures. The observed values of f l ~ ~ are again much smaller than PAA, to an extent which is unlikely to be due to the simple effect of mass or inter- molecular potential.The energy-level diagrams again show easy vibration- vibration transfers, and the simplest hypothesis is that the same mechanism applies as for the C2F4 mixtures discussed above. Table 1 shows values of PtB calculated on the same basis, and their ratios to the minimum likely value of pBF. It will be seen that the ratios again approximate to unity, and, since the value of p”,” is only approximate and it is unlikely that PTA will be much larger or smaller, this can be regarded as satisfactory agreement. It is, of course, possible that the self-deactiva- tion of CH30CH3 (process (4)) is faster than the complex transfer, so that the latter is rate-determining. But, in view of the fact that the lowest fundamental of CH30CH3, 164 cm-1, is considerably larger than Av for any of the complex col- lisions involved, this seems unlikely, and the mechanism proposed above has the merit of giving a self-consistent interpretation of the results for both sets of mixtures.An attempted interpretation on the assumption that the complex collision is rate- determining leads to mutually contradictory results for some of the mixtures, and there is no obvious physical reason for applying it in some cases and not in others. The authors are grateful to Dr. C. J. Danby for making mass spectrometer analyses of mixtures, to Dr. J. L. Stewart for help on constructional details of the 4 Mclsec interferometer and to I.C.I. Plastics Division for generously providing the tetrafluoroethylene used.A. J. E. and D. P. are indebted to the Department of Scientific and Industrial Research for maintenance allowances, and J. L. S. to the Central Electricity Generating Board for support. 1 Garvin, Broida and Kistiakowsky, J. Chem. Physics, 1960, 32, 880. 2 Sette, Busala and Hubbard, J. Chem. Physics, 1955, 23, 787. 3 Lambert and Salter, Proc. Roy. SOC. A , 1957, 243, 78. 4 Herzfeld and Litovitz, Absorption and Dispersion a f Ultrasonic Waues (Academic Press, New 5 Tuesday and Boudart, Tech. Note 7, Contract AF 33(038)-23976 (Princeton University, 1955). 6 Amme and Legvold, J . Chem. Physics, 1957, 26, 514. 7 Calvert and Amme, J . Chem. Physics, 1960, 33, 1270. 8 Lambert and Salter, Proc. Roy. SOC. A , 1959, 253, 277. 9 Arnold, McCoubrey and Ubbelohde, Proc. Roy. Soc. A , 1958,248,445. 10 Danby, Lambert and Mitchell, Proc. Roy. Soc. A , 1957, 239, 365. 11 Hirschfelder, Curtiss and Bird, Molecular Theory of Gases snd Liquids (Wiley, New York, York, 1959). 1954).70 VIBRATIONAL ENERGY TRANSFER 12 Guggenheim and McGlashan, Proc. Roy. SOC. A, 1951, 206,448. 13 Eucken and Franck, 2. Elektrochem., 1948, 52, 195. 14 Kistiakowsky and Rice, J. Chem. Physics, 1940, 8, 618. 15 Arnett and Crawford, J. Chem. Physics, 1950, 18, 118. 16 Hamann, McManamey and Pearse, Trans. Faraday SOC., 1953, 49, 351. 17 Clegg, Rowlinson and Sutton, Trans. Faraday Soc., 1955, 51, 1327. 18 Lagemann and Jones, J. Chem. Physics, 1951, 19, 534. 19 Fogg, Hanks and Lambert, Proc. Roy. SOC. A, 1953,219,490. 20 Dr. E. B. Smith, private communication. 21 Plyler and Benedict, J. Res. Nat. Bur. Stand., 1951, 47, 202. 22 Hamann and Lambert, Austral. J. Chem., 1954, 7, 1. 23 Herzberg, Infa-red and Raman Spectra of PoIyatomic Molecules (van Nostrand, New York, 24 Fogg and Lambert, Proc. Roy. SOC. A, 1955, 232, 537. 25 Mann and Plyler, J. Chem. Physics, 1955, 23, 1989. 26 Mashiko, Tokyo Govt. Chem. Znd. Res. Inst., 1958, 53, 162. 27 Whytlaw-Gray, Reeves and Bottomley, Nature, 1958, 181, 1004. 28 Hamann and Pearse, Trans. Faraday Soc., 1952, 48, 101. 29 Hirschfelder, McClure and Weeks, J. Chenz. Physics, 1942, 10, 201. 30 Claasen, J. Chem. Physics, 1954, 22, 50. 31 Edmonds and Lamb, Proc. Physic. Soc., 1958, 71, 17. 3.2 Cottrell and McCoubrey, Molecular Energy Transfer in Gases (Butterworths, London, 1961), 1945). p. 28.

ISSN:0366-9033    

DOI:10.1039/DF9623300061

出版商:RSC    

年代:1962

数据来源: RSC

摘要:

Rotational Energy Transfer in Molecular Transitions in Parahydrogen Collisions : BY W. D. DAVISON Dept. of Theoretical Chemistry, University Chemical Laboratory, Lensfield Road, Cambridge Received 13th December, 1961 The theory of rotational transitions in the collision of two diatomic molecules is outlined, and the effect of quantum statistics discussed. A general expression is obtained for the matrix elements of the angular terms appearing in a non-spherical potential function. The full distorted wave method is employed to calculate the cross-sections for the 0,O +2,0 and 2,0 +4,0 rotational transitions in parahydrogen, the anisotropy of a " exp-six " potential being varied to secure agreement with ultrasonic dispersion measurements. ~~~~ ~ 1. INTRODUCTION In the collisions of light diatomic molecules in the gas phase, the inefficient transfer of rotational energy gives rise to the well-known phenomena of the rotational absorption and dispersion of ultrasonic waves.1 The rotational relaxation effects behind shock fronts2 and in thermal conductivity measurements 3 are also of current interest.The theoretical treatment of such problems has received much attention recently ; in particular, the cross-sections for transitions between the first few rotational levels in parahydrogen have been calculated with reasonable success by Beckerle,4 Brout 5 and Takayanagi,6 the latter author providing the most com- prehensive discussion. The chief source of uncertainty in such work is our limited knowledge of the details of the intermolecular potential function ; on the other hand, the many practical difficulties which arise in a detailed treatment have led previous workers to introduce several simplifying approximations.A more accurate analysis is here described. n 0 FIG. 1 .-Collision of two diatomic molecules. 2. COLLISION THEORY For the experimental situations in which we are interested, it is an excellent approximation to regard a diatomic molecule as a rigid rotator. The theory of 7172 ROTATIONAL ENERGY TRANSFER the scattering of an atom or ion by such a molecule has been examined in detail by Arthurs and Dalgarno,' and we first outline the straightforward extension of their treatment to the inelastic collision of two rigid rotators. Although we shall not fully exploit this more general formulation in the present work, it is required for other calculations ; for example, the estimation of the cross-section for simul- taneous transitions in the collision of polar molecules.For the moment we suppose that the two molecules are distinguishable (cf. 5 3 below). Atomic units are used throughout. The configuration of the system is described with respect to a set of space-fixed axes as indicated in fig. 1 ; GI, G 2 are the centres of mass of the two molecules, and we shall use the abbreviation f l = (61,41), etc. We make the following definitions : of the ith molecule I rotational Hamiltonian reduced mass moment of inertia rotational angular momentum quantum numbers reduced mass of the whole system, intermolecular potential, Laplacian for the relative motion of the two molecules, relative translational energy before the collision, quantum numbers for the resultant rotational angular momentum of the two molecules (j12 = j1+ j2), orbital (or relative) angular momentum, total angular momentum quantum numbers (J = j12+1).It is convenient to denote the frequently recurring groups of indices jl j2 j12Z and j2jlj121 by the single symbols n and ii respectively. If the molecules are initially in rotational states j1, ml and j2, m 2 , any wave function Y( j1,j2 I R,?& describing the complete system must satisfy the wave equation where and is the total energy of the system, the motion of its centre of mass having been separ- ated out in the usual way. We expand Y(j1, j2) in terms of the basic set of functions, CV~M(~~~P,,~J = C C ( j121mM-m I JW x + ~ I Z +it (4) m = - j l z r n l s - j l (jljzmlm - ml1 jlzni) q M -m&> Yjlml(il> Yj2m-ml(iz)y in which the rigorously good quantum numbers Jand M appear explicitly ; (abcrp I cy) is the Wigner vector-cou]i.ling coefficient, and the Y's are normalized spherical har- monics.As is usual in the stationary-state formulation of collision theory which we have adopted here, our ultimate aim is to obtain the solution of (1) which, for large values of R, represents an incoming plane wave carrying the initial rotational states, and outgoing spherical waves associated with various possible final states. First. however. it is convenient to consider the less general situation in which J, M ,W. D. DAVISON 73 j12 and I are sharply defined before the collision. Noting that J and M will remain unchanged throughout, the wave function for the system may then be written Y,"'(R,f,,f,) = CR- 'u$(R>Yi?(&,,i2).( 5 ) n' In order that (5) should represent the scattering of an incoming spherical wave, the radial functions are required to have the asymptotic form ui!'(R) - exp [ - i ( kI2R-- ; ) ] - ( z y S J ( n ; n ' ) exp R-+W SJ(n;n') is an element of the complex scattering matrix, the knowledge of which is sufficient for a complete description of the collision. Substitution of (5) in (1) yields the set of coupled equations where A (n'J I Y I n"J> = Yi?'VY,J.fndRdf,dt2 SSS and the wave numbers before and after a jl,jz+ji, j i transition, k12 and ki2 = k(j1,j2;ji, j ; ) , are given by We note that the matrix elements (8), as well as being diagonal in J and M, are also independent of M, since Vis invariant under rotation of the co-ordinate axes.To investigate the scattering of a pZane wave carrying the initial states and in- cident along the axis @ = 0, we combine the spherical waves (5) to give Q(j1,j2) = exp (ik,,R cos @)Y)12! Yj;;: (10) (jI2ZmO I JM)( j , j2mlm2 I j12m)( j;2Zrnz'M - m' I JM)( j ; j;m;mi I ji2m') x where TJ(n;nl) = Sn,nt - SJ(n;n!).74 ROTATIONAL ENERGY TRANSFER The total cross-section for the j l , j z + j i , jb transition is then given in terms of the scattering amplitudes by c C, J’ I 4(j1j2mIm2; j; iim;mk I*R) 1 2 ; ~ ~ (15) m l m ml,m2 where we have averaged over ml,m2, and summed over m;,rnL. (This procedure leads to an important simplification in subsequent expressions for the cross-section, but is, of course, only valid for an assembly of molecules under conditions such that there is a uniform distribution of molecules amongst the magnetic substates associ- ated with a given angular momentum.) After considerable analysis, in the course of which the summations over the magnetic quantum numbers are eliminated by an extension of the work of Blatt and Biedenharn,g (1 5) is finally reduced to The method used for the approximate evaluation of (16) depends on the specific problem under investigation. For parahydrogen, the change in relative translational energy during a rotational transition is large, while the departure from spherical symmetry is small, so that the non-diagonal matrix elements are much smaller than the diagonal elements ; the distorted wave approximation is then appropriate.We take as a first estimate to the solution of (7), u$ = ann,u$, (17) where o$ is the solution of the elastic scattering equation, which has the asymptotic form The phase shifts ~2 provide a measure of the distortion of the spherical waves by the potential. Inserting (17) in (7), we obtain an improved solution 9 with the asymptotic form, where Obtaining the scattering matrix by comparison with (6) and substituting in (16), the distorted wave approximation to the total inelastic cross-section is found to beW. D. DAVISON 75 The distorted wave approximation is essentially a first-order perturbation pro- cedure, based on a zero-order situation in which all non-diagonal matrix elements of the potential are neglected, so that only elastic scattering under the diagonal matrix elements is possible.The partial wave expansion of the familiar Born approxim- ation is obtained if in addition we neglect the diagonal matrix elements-in the classical limit this corresponds to replacing the actual curved orbit by a straight line path (the method of impact parameters). In most cases, however, the Born approximation leads to a gross overestimation of the cross-section, since the dis- tortion is much too great to be neglected in this way. This is because the inelastic scattering is controlled principally by the strong repulsive interactions at small separations, so that, of the partial waves which contribute significantly to the in- elastic cross-section, a large proportion have associated phase shifts of the order of several radians.A notable exception is provided by the pressure broadening of microwave spectral lines ; this can be explained in terms of the rotational transitions induced by long-range interactions, the diagonal matrix elements of which are often zero. 3. THE COLLISION OF IDENTICAL MOLECULES When the molecules are of the same species, a complete discussion should take account of the modifications in the theory which are required by quantum statistics. For Bo s e-Eins t ein stat istics (e. g . , parahydro gen and other homo nuclear molecules) , any wave function describing the complete system must be unchanged by the trans- formation (R,rl,r2)-+( - R,rz,rl), corresponding to interchange of the two molec- ules; in the Fermi-Dirac case, such a function is changed in sign only.For sim- plicity we shall only consider the collision of homonuclear molecules with zero resultant nuclear spin; the more general situation has been discussed by Kerner.10 To facilitate comparison with the " unsymmetrized " theory of the previous section, it is convenient to retain the expansion (5) of Yi". We write ~ ' = j ; + j ; - - j ; ~ + l ' . Then, since the symmetry requirement introduces the restriction u,JI = (- 1y'u;lr: (24) except when j ; = j)2 and I' - j ; , is odd, in which case u$ is zero. We proceed with the analysis as before, bearing in mind that the order of j ; , j.; in the scattering matrix has now no significance (corresponding to the fact that it is no longer possible to " label " the molecules).We use a result of Gioumousis and Curtiss 11 to obtain the final expression for the " symmetrized " cross-section. In the distorted wave approximation this takes the form * where * Tildes are used to denote the " symmetrized " form of quantities previously defined in 5 2.76 ROTATIONAL ENERGY TRANSFER * ad? has the asymptotic form (19), as before, but is now the solution of (- l)”(G’J 1 V 1 dJ>} u:!’ = 0, (27) 1 where The cross-section (25) refeis to a collision in which two molecules, initially in states jI,j2, are finally in states j ; , j i , either the transition jl+ji,jz+ji or the transition jl+j;,j2+j having occurred. In many cases the exchange terms in (27) are small, so that ~ ~ ~ ~ c I I $ and (const.)x (- 1>.’(2J+ l)fl;!’&!! (30) Provided that a sufficient number of partial waves contribute significantly to the cross-section, the last term of (30) is negligible ; then J 1J’ j 1 2 , j i i and the effect of quantum statistics vanishes in the classical limit, as expected.Simultaneous transitions in both molecules have not been considered in the present calculations, so it is convenient to express the results in terms of the cross- section, Y ry Q(jl ,j2+ i A) = , j 2 ) U , j 2 Q f h ,h-4 , . j 2 ~ Q ( . i 1 J 2 -+j; ,L>. 4. MATRIX ELEMENTS OF ANGULAR-DEPENDENT POTENTIALS The potential between two rigid diatomic molecules may be expanded in terms of Legendre polynomials : ~ ( ~ ; 1 9 ; 2 ) = C urst(R)pr(cos X I 2 P s ( c o s )P~(cosx~>, (32) r,s,t where ~ 1 2 is the angle between rl and r2.At present, it is not possible to give more than rather rough estimates of the first few radial factors v,,t(R), and it is recognized that quite small variations in these functions can lead to marked changes in the calculated cross-sections. It is to be hoped, however, that following the recent progress in the ab initio calculation of molecular properties, it may soon be possible to carry out accurate investigations of the angular dependence of the potential between some simple systems. We therefore seek a general expression for the matrix elements of the angular factors in (32). The most compact formulation is in terms of the tensor algebraW. D. DAVISON 77 of Fano and Racah; 12 reference should be made to this book for a discussion of the notation employed.Expanding in spherical harmonic tensors, we have : s + t CAI Is1 Ctl CAI = c u [C(E)xC(Z)] I= I s-t I (34) where C(Z) denotes the spherical harmonic tensor associated with k, (abc)o is a contraction for the coefficient (abOO 1 cO), and CAI Crl Crl Is1 [tl [A1 u = (C( 1) C(2))[C( 1) x C(2)] . Then (n'J I Pr(cos xl2)Ps(cos xl)P,(cos x2) 1 n J ) = xi"+'-' (- 1)J+"+j12 (st40 x il It may be shown that r + s r + t [a1 IS1 CAI - W(upA/tsr)[C(l) x C(2)] . (38) Substituting in (37) and evaluating the reduced matrix elements, we deduce that {n'J 1 P,(cos xl2)Ps(cos xl)P,(cos x2) I n J ) = (- 1)J+j12+ji+ji+r~ a,BJ (rsa>o(rtP>o(st4o( j i j al0(ji j2P) o(l' Wo x [(2j1 + 1)(2j; + 1)(2j2 + 1)(2j; + 1)(2jI2 + 1)(2ji2 + 1)(2Z+ 1)(21'+ I)]* x - w(uPL/tsr)%ji 2 j 1 ~ W J ) X C ~ L ~ l a ~ j i j2~/ji2.il (39) For any given problem, the matrix elements (including, if necessary, those of multipole interactions) can be straightforwardly derived as particular cases of (39) ; when a zero appears in any coefficient, the indices can always be permuted so as to make use of such relations as (aOb)o = 8ab, - W(abc/deO) = (- l)"+b+c6a,6bd[(2a + 1)(2b + 1)]-+, X(abc/def/ghO) = (- l)b+c+d+g6cf6gh[(2f + 1)(2g + l)]-*i%(abc/edg).(40) For the present calculations we iequire only (n'J 1 Ps(cos xl) I n J ) = (- 1)J+i2+s6j,ii(2s+ l)-'( j; jls)o(lfZs)o x [(2j1 + 1)(2j; + 1)(2j12 + 1)(2ji2 + 1)(2Z+ 1)(2Z'+ I)]+ x - W( j i 2 j 2s1 u J>WC j i z j sij j i j 2). (41) The Racah W-coefficients have been tabulated for limited ranges of the indices only, but can be computed using finite series expressions.78 ROTATIONAL ENERGY TRANSFER 5.ROTATIONAL TRANSITIONS I N PARAHYDROGEN The collisional excitation of the first vibrational level in hydrogen has been studied by Salkoff and Bauer,l3 these authors being the first to employ the distorted wave method without approximation in such a problem. As a simple application of the foregoing theory, we have carried out an analogous investigation of the rotational problem, confining ourselves to the 0,0+2,0 and 2,0+4,0 transitions in para- hydrogen. The numerical work was performed on the EDSAC 2 computer in the University Mathematical Laboratory, Cambridge. (i) H2-H2 POTENTIAL We have used two simple representations of the potential between two hydrogen molecules.The first has the Morse form V(R,t1,i2) = A exp [ - 2a(R - R,)] - 2A exp [ - cc(R - R,)] + PA exp c - 24R - Ro)IP2(cos Xl) + mcos X2)J (42) where A = 1.1 x 10-4, a = 0.935, Ro = 6.4 and B = 0.075. This potential has already been used by Takayanagi ; 6 the repulsive terms provide a reasonable fit to the theoretical calculations of Evett and Margenau 14 when higher-order terms in the expansion (32) are neglected. In Evett and Margenau’s work, however, the approximation of certain of the rnulticentre integrals reduces the predicted anisotropy of the potential, while in the subsequent calculations of Mason and Hirschfelder 15 there are other approximations which lead to an abnormally high angular dependence -corresponding to PeO-4 in (42).The “ shape ” of the hydrogen molecule is certainly not far from spherical, and we might reasonably expect B to lie in the range 0.1 to 0.2. In the second potential used in our calculations, we have to some extent allowed for this uncertainty in the short-range potential terms, at the same time taking more careful account of the long-range attractive forces. We use the “ exp-six ” function V(R,t1,+2) = A exp [ - 2a(R - R,)] - BR-6 + (PA exp [ - 2a(R - R,)] - DR+) x where A, a and Ro have the same values as before, B = 11.0, D = 0.8 and p is re- garded as a parameter to be varied. The long-range R-6 terms are based on the calculations of Britton and Bean.16 The quadrupole-quadrupole interaction and other higher order terms have been omitted, their matrix elements being very small or zero for the rotational transitions considered in this paper.These terms are, of course, important in simultaneous transitions in the two molecules, but con- sideration of these transitions must be deferred until the higher-order terms in the short-range potential are known with some accuracy. [PZ(COS Xl) +P,(cos X2)IY (43) (ii) NUMERICAL INTEGRATION The Adams method 17 was employed for the numerical integration of the radial eqn. (27). The potential was taken as effectively infinite for R<2-0, so that the integration of each equation was begun at R = 2.0 with the initial value of the solu- tion zero and with a suitably small initial gradient, and continued outwards to R = 15.0, beyond which point the potential is quite negligible.The integration interval was given its smallest value at small R (where the potential is varying rapidly), and was increased twice as the integration proceeded outwards; it was, however, kept sufficiently small over the whole range so that at higher impact energies theW. D. DAVISON 79 rapid oscillations in the solutions were accurately reproduced. The phase shifts and normalization constants for each solution were found by comparing the values of the solution at R1 = 14.5 and R2 = 15.0 with the values of the corresponding spherical Bessel functions at these points. Thus, if a solution with the asymptotic form, and The spherical Bessel functions required were generated without appreciable error by means of the standard recurrence relation.(iii) c A L c u LA TIONS In our first calculations, the cross-section &0,0-+2,0) was evaluated using the potential function (42). J is restricted by symmetry to even values only; the four radial equations for each value of J (corresponding to I’ = J+2,J,J-2 and I = J ) were integrated simultaneously. The required angular matrix elements were first computed and then combined with the appropriate radial factors at each step of the integration to form the diagonal and coupling matrix elements. On completion of the integration, the contributions to the cross-section from the current value of J were printed, together with details of the solutions. The calculations were then repeated for further values of J until the contributions to the cross-section became negligibly small.The computed cross-sections for several incident energies are included in table 1. TABLE 1 .-CROSS-SECTIONS FOR ROTATIONAL TRANSITIONS IN PARAHYDROGEN (a;). incident wave number, kl2 244 2-60 2.80 3.10 3.50 3.73 400 4-50 5.00 5-50 6.25 T = 197.1”K <Q,y> { T = 298.4”K e” (0,@+2,0; kl2) Morse exp-six (16 = 0-14) 0~0oO0 0*0000 0.0362 00551 0.0755 0.1335 0.1508 0.2964 0.27 13 0.583 1 04416 1.0146 0.8118 2.0416 - 3.5324 3.3x 10-2 6 . 8 ~ 10-2 2.2 x 10-1 - - - - - - - 0~0000 0 * 0 0 5 0 2 0.0374 0.1121 0.2331 0.5006 2.2 x 10-4 2 . 8 ~ 10-3 When we come to consider transitions involving states of higher angular momentum, we are faced with the practical difficulty of integrating a much larger number of radial equations for each value of J.The consequent increase in com- puter time is hardly worthwhile in view of the rather rough potential functions available. If, however, in the radial eqn. (27), we neglect angular distortion (re- placing the diagonal matrix elements by the spherical part of the potential), the80 ROTATIONAL ENERGY TRANSFER number of equations to be solved is greatly reduced. Provided that the departure from spherical symmetry is small, the error in the final cross-section introduced by this approximation is not serious; some sample calculations have indicated that in the present case it is at the most a few pzr cent. Following this simplified procedure, we have evaluated the cross-sections Q(0,0--+2,0) and 5(2,0-+4,0) using the potential function (43). It was found convenient to carry out the calculations in two stages.The distorted waves were first computed for many values of J, and transferred to permanent storage. In the second part of the programme, they were called down to the fast store as required, and integrated together with the appropriate angular matrix elements to form the integrals (26). This scheme was made sufficiently general (especially with regard to the evaluation of the Racah coefficients) that, given the relevant angular terms in the potential function, the cross-sections for other transitions could be computed with equal facility. Furthermore, the programme was so devised that the cross- section corresponding to any desired values of the potential parameters /? and D could be easily deduced from the final output.The variation of a and A exp (2aRo) Y J FIG. 2.-Partial cross-sections as a function of J. (a) Z(0,O +2,0 ; 2.60) X 10, (b) Z(0,O +2,0 ; 6*25), (c) Z(2,O +4,0 ; 4.50) X 102, ( d ) &2,0 +4,0 ; 5.50)X 10. Note that contributions to -&0,0-+2,0) come from even values of J only. is not so straightforward and was not attempted. For a potential of the form (42), the cross-section (neglecting angular distortion) is proportional to /?2 ; with the in- clusion of a long-range angular term in (43), the cross-section is considerably reduced for a given value of 2, particularly near the threshold, and is somewhat more sensitive to variation of p. The cross-sections corresponding to p = 0.14 lead to satisfactory agreement with experiment (see below), and are included in table 1.In fig. 2, the contributions to the cross-sections from individual values of J have been plotted against J for some typical energies. It will be seen that at the highest energies con- sidered there are significant contributions from values of J up to Jfi30.W. D . DAVISON 81 (iv) COMPARISON WITH EXPERIMENT In correlating our results with the dispersion measurements of Rhodes,ls we assume that the probability of a molecule making a transition is independent of the rotational state of its collision partner (in particular, we ignore the possibility of simultaneous transitions). This assumption is consistent with the simple form we have adopted for the potential function, and allows us to interpret the rotational dispersion in terms of a simple three-level mechanism.The deviations from perfect gas behaviour are sufficiently small to be neglected. incident wave number, k12 FIG. 3.-Averaging over the Maxwellian distribution at 197.1 O K . (4 M(kl2)Y (6) G(0,O +2,0 ; 4 2 ) x 10-1, (4 Z(2,04,0; k12), (4 M(k12) x e'C0,o +2,0 ; k12) x 10, (e) M(k12) x S(2,O +4,0 ; k12) x 103. It may then be shown that the probability per second that a molecule in the rotational state j will make a transition to the state j' is where p is the pressure, k the Boltzmann constant, and ( Q j y ) the cross-section for the j-j' transition averaged over the Maxwellian distribution of incident wave numbers ; < Q j j * ) = F ( k 1 2 ) i i ~ , o + j i o ;k, 2 ~ 1 2 ; Fig. 3 illustrates the formation of the integmnd in (48).82 ROTATIONAL ENERGY TRANSFER Solving the relaxation equations for a periodic disturbance of frequency 0/2n.we obtain the following expression for the effective rotational specific heat per mole at temperature T : E2, E4 are the energies of the j = 2,4 levels respectively, and ni is the number of molecules per mole in the j = 2 level at equilibrium. The ultrasonic dispersion curve may be calculated from the expression for the complex sound velocity V,. As expected, the Morse potential (which was considered chiefly for the sake of comparison with Takayanagi's less rigorous cal- culations) yields a value for ((302) which is clearly too low. If we choose p = 0.14 in the potential (43), the calculated dispersion curve is in satisfactory agreement with Rhodes' measurements at 197-1"K (see fig.4). At room temperature, however, the theoretical curve (with the same value of p) is too high at lower frequencies. I I 1 1 I I I f I 1 1 1 I I I I I I 10 100 4 2 n p (Mclatrn) FIG. 4.-Rotational dispersion in parahydrogen ; expt. points (Rhodes) : x , 197-1°K ; 0, 298.4"K. This can be partly attributed to the limitations of the potential (43), especially the assumption that the spherical and angular terms have the same exponential de- pendence on R. However, as pointed out by Takayanagi,lg the discrepancy probably arises for the most part from the neglect of simultaneous transitions, particularly the 2,2++4,0 transition, where the change in relative translational energy is com- paratively small. 6. DISCUSSION Takayanagi has made extensive use of the " modified wave number " approxim- ation, in which the centrifugal potential Z'(Z' + l)/R2 in the radial equations is replacedW.D. DAVISON 83 by its value at a distance R,, chosen to be of the order of the distance of closest approach in a typical collision. The problem is thereby reduced to that of a one- dimensional collision with effective wave number kLff given by if the s-wave (I’ = 0) radial equation can be solved in closed form, the direct numerical solution of the radial equations is avoided, and considerable simplification results. This procedure (with some refinements) has proved valuable in exploratory cal- culations, for vibrational as well as rotational transitions, but has two obvious drawbacks. First, it is useful only within the very restricted range of potentials for which an analytic solution of the s-wave equation is available (thus, it is of no help in calculations with the potential (43)).Secondly, the arbitrariness of the parameter R, introduces an added uncertainty into the final cross-sections, in con- trast to an accurate numerical solution, where the adoption of a given potential function leads to unambiguous values for the cross-sections. For the present problem, Takayanagi 6 has used the Morse potential (42) and obtained cross-sections proportional to x = p2Rz ; R, was chosen so as to provide the best fit to expe. I rment at lower temperatures.19 The final dispersion curves are very similar to those in fig. 4. This semi-empirical procedure gives us little information as to the “ best ” values of p and R, ; the former quantity is of interest in that it provides a measure of the actual anisotropy of the potential, the latter in that it might serve as a guide in further applications of the modified wave number approximation, particularly in cases where comparison with experiment is not yet possible.The present calculations were undertaken in the hope of securing closer agree- ment with experiment at room temperature. In fact, although there are differences in detail which are only to be expected, our final results are in overall agreement with those of Takayanagi, and it would appear that no further improvement is possible within the limitations imposed by the potentials (42) and (43). In retrospect, the chief value of our investigation would seem to lie in the fact that, having successfully overcome many of the computational difficulties associated with this kind of work (for example, the accurate generation and manipulation of a large number of distorted waves, and the evaluation of the matrix elements), we can be confident that the wave eqn.(1) has been solved with reasonable accuracy, and that defects in the potential function are therefore responsible for most of the remaining discrepancy between theory and ex@riment. It is clear that accurate calculations of this kind may soon provide an important link between the theoretical prediction of inter- molecular potentials and, for example, the direct measurement of individual cross- sections by molecular beam techniques. For rotational transitions in the collision of heavier molecules, the effect of higher-order coupling in the eqn.(7) cannot be neglected. For vibrational transi- tions, on the other hand, the distorted wave procedure is probably adequate in most cases. As an outgrowth of this work, programmes to treat such problems are at present being developed. I am grateful to Prof. H. C. Longuet-Higgins, F.R.S., for his help and encourage- ment throughout this work, and to Prof. A. Dalgarno and Dr. A. M. Arthurs for some valuable discussions. I also wish to express my thanks to the Director and Staff of the University Mathematical Laboratory, Cambridge, for the generous facilities afforded to me on EDSAC 2, and to Gonville and Caius College for the award of a Rhondda Studentship.84 ROTATIONAL ENERGY TRANSFER 1 Herzfeld and Litovitz, Absorption and Dispersion of Ultrasonic Waves (Academic Press, New 2 Andersen and Hornig, Mol. Physics, 1959, 2,49. 3 Srivastava and Barua, Proc. Physic. SOC., 1961, 77, 677. 4 Beckerle, J. Chem. Physics, 1953, 21, 2034. 5 Brout, J. Chem. Physics, 1954, 22, 934. 6 Takayanagi, Proc. Physic. SOC. A, 1957, 70, 348 ; Sci. Reports Saitama Univ. A, 1959, 3, 37 ; 7 Arthurs and Dalgarno, Proc. Roy. Sac. A, 1960, 256, 540. 8 Blatt and Biedenharn, Rev. Mod. Physics, 1952, 24, 258. 9 Mott and Massey, The Theory of Atomic CoZIisions (Clarendon Press, Oxford, 1949). 10 Kerner, Physic. Rev., 1953, 91, 1174. 11 Gioumousis and Curtiss, J. Chem. Physics, 1958, 29, 996. 12 Fano and Racah, Irreducible TensoriaZ Sets (Academic Press, New York, 1959) ; see also 13 Salkoff and Bauer, J. Chem. Physics, 1958, 29, 26. 14 Evett and Margenau, Physic. Rev., 1953, 90, 1021. 15 Mason and Hirschfelder, J. Chem. Physics, 1957, 26, 756. 16 Britton and Bean, Can. J. Physics, 1955, 33, 668. 17 see, for example, Booth, Numerical Methods (Butterworths, London, 1955). 18 Rhodes, Physic. Rev., 1946, 70, 932. 19 Takayanagi, J. Physic. Soc. Japan, 1959, 14, 1458. York, 1959). and further references therefrom. Edmonds, Angular Momentum in Quantum Mechanics (University Press, Princeton, 1957).

ISSN:0366-9033    

DOI:10.1039/DF9623300071

出版商:RSC    

年代:1962

数据来源: RSC